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p. 5, line 20, delete "\(\pi\)."
p. 34, line 20, for "\(yRx\)" read "\(xRy\)."
p. 36, line 7 and line 10, for "\(Q\mid R\)" read "\(R\mid P\)."
p. 44, line 17, for "\((p).p. \text{is false}\)" read "\((p). p\text{ is false}\)."
p. 112, in *2·52, in place of "\(p \supset \sim q\)" read "\(\sim p \supset \sim q\)."
p. 129, in *5·11, in place of reference to "\(*2·51\)" read reference to "\(*2·5\)."
p. 129, in *5·12, in place of reference to "\(*2·52\)" read reference to "*2·51."
p. 144, *10·23 should be "\(\vdash\colon\ldotp(x).\phi x \supset p.\equiv :(\exists x).\phi x.\supset.p\)."
p. 157, line 11, for "*10" read "*9."
p. 184, last line of Dem. of *14·111, for second "\(x=c\)" read "\(x=b\)."
p. 228, in *23·81, for "\(\dot{-} R \unicode{x2abd} \dot{-} S\)" read "\(\dot{-} S \unicode{x2abd} \dot{-} R\)."
p. 242, in *25·37, for "\(zRw\)" read "\(zSw\)."
p. 242, in *25·412, for "\(R\)" read "\(S\)."
p. 253, 2nd and 4th lines of Dem. of *31·16, for "*21·35" read "*23·35."
p. 259, in note to *32·35, for "*32·2" read "*32·3."
p. 263, in *33·16, 4th line of Dem., for "*20·34" read "*22·34."
p. 265, in *33·26, 2nd line of Dem., for "*21·34" read "*23·34."
p. 275, in *34·6, 4th line of Dem., for first "\(S\)" read "\(R\)."
p. 289, 1st line, for "\(=\beta \uparrow \gamma\)" read "\(=\alpha \uparrow \gamma\)."
p. 322, in *40·18, enunciation, for "\(\equiv\)" read "\(=\)".
p. 329, in *40·69, Dem., for "\(\overrightarrow{P}\)" read "\(\overleftarrow{P}\)" (3 times).
p. 387, in *55·224, 1st line of Dem., for "\(\uparrow\)" read "\(\downarrow\)" (twice).
p. 388, in *55·281, for third "\(=\)" read "\(\equiv\)."
p. 410, in *60·53, last line of Dem., for "\(\gamma\)" read "\(\beta\)."
p. 453, in *71·25, Dem., 1st line, for "\(xRy.xRz\)" read "\(ySx.zSx\)."
"        "                              2nd line, for "\(xRy.ySu.xRz.zSv.\supset.y=z.ySu.zSv\)"
read "\(uRy.ySx.vRz.zSx.\supset.y=z.uRy.vRz\)."
"        "                              3rd line, for "\(ySu.ySv\)" read "\(uRy.vRy\)."
"        "                              6th line, for "\(xRy.ySu\)" read "\(uRy.ySx\)" and for
"\(xRz.zSv\)" read "\(vRz.zSx\)."
"        "                              7th line, for "\(x(R\mid S)u.x(R\mid S)v\)" read "\(u(R\mid S)x.v(R\mid S)x\)."
p. 465, in *72·16, Dem., 1st line, for last "\(\kappa\)" read "\(x\)."
p. 483, in *73·44, Dem., 1st line, for second "\(y\)" read "\(x\)."
p. 485, in *73·511, for "\(\beta\)" read "\(\alpha\)."
p. 522, in *81·23, enunciation and 2nd line of Dem., for "\(\overrightarrow{R}\)" read "\(R\)."
p. 592, in *91·33, Dem., 1st line, for "\(P\)" read "\(R\)."
p. 614, in *93·36, Dem., for "\(R\)" read "\(P\)" throughout.
p. 628, in *95·21, Dem., line 6, for "\(Q'\)" read "\(T'\)."

ERRATA TO VOLUME II

p. 82, last line but one, for "\(\Lambda_{\alpha}\)" and "\(\Lambda_{\beta}\)" read "\(\Lambda \cap \alpha\)" and "\(\Lambda \cap \beta\)."
p. 101, *112·23, enunciation, the second time two dots occur, read one dot.
p. 573, *205·7, enunciation, for "\(\text{max}_{P}\)" read "\(\overrightarrow{\text{max}}_{P}\)."

THE purpose of the following observations is to bring together in one discussion various explanations which are required in applying the theory of types to cardinal arithmetic. It is convenient to collect these observations, since otherwise their dispersion throughout the several numbers of Part III makes it difficult to see what is their total effect. But although we have placed these observations at the beginning, they are to be read concurrently with the text of Part III, at least with so much of the text as consists of explanations of definitions. The earlier portion of what follows is merely a résumé of previous explanations; it is only in the later portions that the application to cardinal arithmetic is made.

I. General Observations on Types.

Three different kinds of typical ambiguity are involved in our propositions, concerning:

(1) the functional hierarchy,

(2) the propositional hierarchy,

(3) the extensional hierarchy.

The relevance of these must be separately considered.

We often speak as though the type represented by small Latin letters were not composed of functions. It is, however, compatible with all we have to say that it should be composed of functions. It is to be observed, further, that, given the number of individuals, there is nothing in our axioms to show how many predicative functions of individuals there are, i.e. their number is not a function of the number of individuals: we only know that their number \(\geq 2^{\text{Nc}ʻ\text{Indiv}}\), where "\(\text{Indiv}\)" stands for the class of individuals.

In practice, we proceed along the extensional hierarchy after the early numbers of the book. If we have started from individuals, the result of this is to exclude functions wholly from our hierarchy; if we have started with functions of a given type, all functions of other types are excluded. Thus a fresh extensional hierarchy, wholly excluding every other, starts from each[Pg x] type of function. When we speak simply of "the extensional hierarchy," we mean the one which starts from individuals.

It is to be observed that when we have the assertion of a propositional function, say "\(\vdash.\phi x\)," the \(x\) must be of some definite type, i.e. we only assert that \(\phi x\) is true whatever \(x\) may be within some one type. Thus e.g. "\(\vdash.x=x\)" does not assert more than that this assertion holds for any \(x\) of a given type. It is true that symbolically the same assertion holds in other types, but other types cannot be included under one assertion-sign, because no variable can travel beyond its type.

The process of rendering the types of variables ambiguous is begun in *9, where we take the first step in regard to the propositional hierarchy. Before *9, our variables are elementary propositions. These are such as contain no apparent variables. Hence the only functions that occur are matrices, and these only occur through their values. The assumption involved in the transition from Section A to Section B (Part I) is that, given "\(\vdash.fp\)," where \(p\) is an elementary proposition, we may substitute for \(p\) "\(\phi!(x,y,z,\ldots)\)," where \(\phi\) is any matrix. Thus instead of "\(\vdash.fp\)," which contained one variable \(p\) of a given type, we have "\(\vdash.f\{\phi!(x,y,z,...)\}\)," which contains several variables of several types (any finite number of variables and types is possible). This assumption involves some rather difficult points. It is to be remembered that no value of \(\phi\) contains \(\phi\) as a constituent, and therefore \(\phi\) is not a constituent of \(fp\) even if \(p\) is a value of \(\phi\). Thus we pass, above, from an assertion containing no function as a constituent to one containing one or more functions as constituents. The assertion "\(\vdash.fp\)" concerns any elementary proposition, whereas "\(\vdash.f\{\phi(x,y,z,\ldots)\}\)" concerns any of a certain set of elementary propositions, namely any of those that are values of \(\phi\). Different types of functions give different sorts of ways of picking out elementary propositions.

Having assumed or proved "\(\vdash.fp\)," where \(p\) is elementary and therefore involves no ambiguity of type, we thus assert \[ \vdash.f\{\phi(x,y,z,\ldots)\}, \] where the types of the arguments and the number of them are wholly arbitrary, except that they must belong to the functional hierarchy including individuals. (The assumption that propositions are incomplete symbols excludes the possibility that the arguments to \(\phi\) are propositions.) The noteworthy point is that we thus obtain an assertion in which there may be any finite number of variables and the variables have unlimited typical ambiguity, from an assertion containing one variable of a perfectly definite type. All this is presupposed before we embark on the propositional hierarchy.

It should be observed that all elementary propositions are values of predicative functions of one individual, i.e. of \(\phi!\hat{x}\), where \(\hat{x}\) is individual.[Pg xi] Thus we need not assume that elementary propositions form a type; we may replace \(p\) by "\(\phi!x\)" in "\(\vdash.fp\)." In this way, propositions as variables wholly disappear.

In extending statements concerning elementary propositions so as formally to apply to first-order propositions, we have to assume afresh the primitive proposition *1·11 (*1·1 is never used), i.e. given "\(\vdash.\phi x\)" and "\(\vdash.\phi x\supset \psi x\)," we have "\(\vdash.\psi x\)" which is practically *9·12. This was asserted in *1·11 for any case in which \(\phi x\) and \(\psi x\) are elementary propositions. There was here already an ambiguity of type, owing to the fact that x need not be an individual, but might be a function of any order. E.g. we might use *1·11 to pass from \[ \unicode{x201c}\vdash.\phi!a\unicode{x201d} \quad \text{and} \quad \unicode{x201c}\vdash.\phi!a\supset \phi!b\unicode{x201d} \quad \text{to} \quad \unicode{x201c}\vdash.\phi!b,\unicode{x201d} \] where \(\phi\) replaces the \(x\) of *1·11, and \(\hat{\phi}!a\), \(\hat{\phi}!b\) replace \(\phi\hat{x}\) and \(\psi\hat{x}\). Thus *1·11, even before its extension in *9, already states a fresh primitive proposition for each fresh type of functions considered. The novelty in *9 is that we allow \(\phi\) and \(\psi\) to contain one apparent variable. This may be of any functional type (including Indiv); thus we get another set of symbolically identical primitive propositions. In passing, as indicated at the end of *9, to more than one apparent variable, we introduce a new batch of primitive propositions with each additional apparent variable.

Similar remarks apply to the other primitive propositions of *9.

What makes the above process legitimate is that nothing in the treatment of functions of order \(n\) presupposes functions of higher order. We can deal with each new type of functions as it arises, without having to take account of the fact that there are later types. From symbolic analogy we "see" that the process can be repeated indefinitely. This possibility rests upon two things:

(1) A fresh interpretation of our constants—\(\lor\), \(\sim\), !, (\(x\))., (\(\exists x\)).—at each fresh stage;

(2) A fresh assumption, symbolically unchanged, of the primitive propositions which we found sufficient at an earlier stage—the possibility of avoiding symbolic change being due to the fresh interpretation of our constants.

The above remarks apply to the axiom of reducibility as well as to our other primitive propositions. If, at any stage, we wish to deal with a class defined by a function of the 30,000th type, we shall have to repeat our arguments and assumptions 30,000 times. But there is still no necessity to speak of the hierarchy as a whole, or to suppose that statements can be made about "all types."

We come now to the extensional hierarchy. This starts from some one point in the functional hierarchy. We usually suppose it to start from[Pg xii] individuals, but any other starting-point is equally legitimate. Whatever type of functions (including \(\text{Indiv}\) we start from, all higher types of functions are excluded from the extensional hierarchy, and also all lower types (if any). Some complications arise here. Suppose we start from \(\text{Indiv}\). Then if \(\phi!\hat{z}\) is any predicative function of individuals, \(\hat{z}(\phi!z) = \phi!\hat{z}\). But identity between a function and a class does not have the usual properties of identity; in fact, though every function is identical with some class, and vice versa, the number of functions is likely to be greater than the number of classes. This is due to the fact that we may have \(\hat{z}(\phi!z)=\psi!\hat{z}.\hat{z}(\phi!z)=\chi!\hat{z}\) without having \(\psi!\hat{z}=\chi!\hat{z}\).

In the extensional hierarchy, we prove the extension from classes to classes of classes, and so on, without fresh primitive propositions (*20, *21). The primitive propositions involved are those concerning the functional hierarchy.

From all these various modes of extension we "see" that whatever can be proved for lower types, whether functional or extensional, can also be proved for higher types[1]. Hence we assume that it is unnecessary to know the types of our variables, though they must always be confined within some one definite type.

Now although everything that can be proved for lower types can be proved for higher types, the converse does not hold. In Vol. I. only two propositions occur which can be proved for higher but not for lower types. These are \(\exists !2\) and \(\exists !2_{r}\). These can be proved for any type except that of individuals. It is to be observed that we do not state that whatever is true for lower types is true for higher types, but only that whatever can be proved for lower types can be proved for higher types. If, for example, \(\text{Nc}ʻ{\text{Indiv}} = \nu\), then this proposition is false for any higher type; but this proposition, \(\text{Nc}ʻ{\text{Indiv}} = \nu\), is one which cannot be proved logically; in fact, it is only ascertainable by a census, not by logic. Thus among the propositions which can be proved by logic, there are some which can only be proved for higher types, but none which can only be proved for lower types.

The propositions which can be proved in some types but not in others all are or depend upon existence-theorems for cardinals. We can prove \[\begin{array}{l} \exists !0,\, \exists !1,\, \text{universally},\\ \exists !2,\, \text{except for}\,\, \text{Indiv},\\ \exists !3,\, \exists !4,\, \text{except for}\,\, \text{Indiv},\, \text{Cl}ʻ\text{Indiv},\, \text{Rl}ʻ\text{Indiv};\,\, \text{and so on}. \end{array} \] Exactly similar remarks would apply to the functional hierarchy. In both cases, the possibility of proving these propositions depends upon the axiom of reducibility and the definition of identity. Suppose there is only one individual, \(x\). Then \(\hat{y}=x\), \(\hat{y}\neq x\) are two different functions, which, by the[Pg xiii] axiom of reducibility, are equivalent to two different predicative functions. Hence there are at least two predicative functions of \(x\), and at least two classes \(\iotaʻx\), \(\Lambda_{x}\). This argument fails both for classes and functions if either we deny the axiom of reducibility or we suppose that there may be two different individuals which agree in all their predicates, i.e. that the definition of identity is misleading.

The statement that what can be proved for lower types can be proved for higher types requires certain limitations, or rather, a more exact formulation. Taking \(\text{Indiv}\) as a primitive idea, put \[ \text{Kl} = \text{Cl}ʻ\text{Indiv} \quad\text{Df},\,\, \text{Kl}^{2} = \text{Cl}ʻ\text{Kl} \quad\text{Df}, \quad\text{etc}. \] Then consider the proposition \(\text{Nc}ʻ\text{Kl} = \Lambda\) . We can prove \[ \text{Nc}ʻ\text{Kl} \cap tʻ\text{Indiv} = \Lambda . \exists ! \text{Nc}ʻ\text{Kl} \cap tʻ\text{Kl} . \exists ! \text{Nc}ʻ\text{Kl} \cap tʻ\text{Kl}^{2}. \quad\text{etc}. \] Thus \(\text{Nc}ʻ\text{Kl} = \Lambda\) can be proved in the lowest type in which it is significant, and disproved in any other. The difficulty, however, is avoided if Indiv is replaced by a variable \(\alpha\), and \(\text{Kl}\) by \(\text{Cl}ʻt_{0}ʻ\alpha\). Then we have \[ \text{Nc}ʻ\text{Cl}ʻt_{0}ʻ\alpha \cap tʻ\alpha = \Lambda, \] and this holds whatever the type of \(\alpha\) may be. Thus in order that our principle about lower and higher types may be true, it is necessary that any relation there may be between two types occurring in a proposition should be preserved; in other words, when one constant type is defined in terms of another (as \(\text{Kl}\) and \(\text{Indiv}\)), the definition must be restored before the type is varied, so that when one type is varied, so is the other. With this proviso, our principle about higher and lower types holds.

With the above proviso, the truth of our statement is manifest. For we have shown that the same primitive propositions, symbolically, which hold for the lowest type concerned in our reasoning, hold also for subsequent types; and therefore all our proofs can be repeated symbolically unchanged.

The importance of this lies in the fact that, when we have proved a proposition for the lowest significant type, we "see" that it holds in any other assigned significant type. Hence every proposition which is proved without the mention of any type is to be regarded as proved for the lowest significant type, and extended by analogy to any other significant type.

By exactly similar considerations we "see" that a proposition which can be proved for some type other than the lowest significant type must hold for any type in the direct descent from this. E.g. suppose we can prove a proposition (such as \(\exists ! 2\)) for the type \(\text{Kl}\) (where \(\text{Kl} = \text{Cl}ʻ\text{Indiv}\)); then merely writing \(\text{Cl}ʻ\text{Indiv}\) for \(\text{Kl}\), we have a proposition which is proved concerning \(\text{Indiv}\), namely \(\exists!2 \cap tʻ\text{Cl}ʻ\text{Indiv}\), and here, by what was said before, \(\text{Indiv}\)) may be replaced by any higher type.

Thus given a typically ambiguous relation \(R\), such that, if \(\tau\) is a type, \(Rʻ\tau\) is a type (\(\text{Cl}\) or \(\text{Rl}\) is such a relation), we "see" that, if we can prove[Pg xiv] \(\phi (Rʻ\text{Indiv})\), we can also prove \(\phi (Rʻ\tau)\), where \(\tau\) is any type, and \(\phi\) is composed of typically ambiguous symbols. Similarly if we can prove \(\phi (\text{Indiv},Rʻ\text{Indiv})\), we can prove \(\phi (\tau ,Rʻ\tau)\), where \(\tau\) is any type. But we cannot in general prove \(\phi (\text{Indiv},Rʻ\tau)\) or \(\phi (\tau,Rʻ\text{Indiv})\), and these may be in fact untrue. e.g. we have \(\exists !\text{Nc}(\text{Kl})ʻ\text{Indiv}.{\sim}\exists!\text{Nc}(\text{Kl})ʻ\text{Kl}^{2}\).

Thus more generally, when a proposition containing several ambiguities can be proved for the types \(Rʻ\text{Indiv}\), \(Sʻ\text{Indiv}\), ..., but not for lower types, it is to be regarded as a function of \(\text{Indiv}\), and then it becomes true for any type; that is, given \[ \phi (Rʻ\text{Indiv},Sʻ\text{Indiv}, \ldots), \] we shall also have \[ \phi (Rʻ\tau,Sʻ\tau, \ldots), \] where \(\tau\) is any type. In this way, all demonstrable propositions are in the first instance about \(\text{Indiv}\), and when so expressed remain true if any other type is substituted for \(\text{Indiv}\).

When a proposition containing typically ambiguous symbols can be proved to be true in the lowest significant type, and we can "see" that symbolically the same proof holds in any other assigned type, we say that the proposition has "permanent truth." (We may also say, loosely, that it is "true in all types.") When a proposition containing typically ambiguous symbols can be proved to be false in the lowest significant type, and we can "see" that it is false in any other assigned type, we say that it has "permanent falsehood." Any other proposition containing typically ambiguous symbols is said to be "fluctuating," or to have "fluctuating truth-value," as opposed to "permanent truth-value," which belongs to propositions that have either permanent truth or permanent falsehood.

In what follows, ambiguities concerned with the propositional hierarchy will be ignored, since they never lead to fluctuating propositions. Thus disjunction and negation and their derivatives will not receive explicit typical determination, but only such typical determination as results from assigning the types of the other typically ambiguous symbols involved.

[Pg xv]

It is convenient to call the symbolic form of a propositional function simply a "symbolic form." Thus, if a symbolic form contains symbols of ambiguous type it represents different propositional functions according as the types of its ambiguous symbols are differently adjusted. The adjustment is of course always limited by the necessity for the preservation of meaning. It is evident that the ideas of "permanent truth-value" and "fluctuating truth-value" apply in reality to symbolic forms and not to propositions or propositional functions. Ambiguity of type can only exist in the process of determination of meaning. When the meaning has been assigned to a symbolic form and a propositional function thereby obtained, all ambiguity of type has vanished.

To "assert a symbolic form" is to assert each of the propositional functions arising for the set of possible typical determinations which are somewhere enumerated. We have in fact enumerated a very limited number of types starting from that of individuals, and we "see" that this process can be indefinitely continued by analogy. The form is always asserted so far as the enumeration has arrived; and this is sufficient for all purposes, since it is essentially impossible to use a type which has not been arrived at by successive enumeration from the lower types.

The only difficulties which arise in Cardinal Arithmetic in connection with the ambiguities of type of the symbols are those which enter through the use of the symbol \(\text{ sm }\), or of the symbol \(\text{Nc}\), which is \(\overrightarrow{\text{ sm }}\). For it may happen that a class in one type has no class similar to it in some lower type (cf. *102·72·73). All fallacious reasoning in cardinal or ordinal arithmetic in connection with types, apart from that due to the mere absence of meaning in symbols, is due to this fact—in other words to the fact that in some types \(\exists!\text{Nc}ʻ\alpha\) is true, and in other types \(\exists !\text{Nc}ʻ\alpha\) may not be true. The fallacy consists in neglecting this latter possibility of the failure of \(\exists !\text{Nc}ʻ\alpha\) for a limited number of types, that is, in taking the "fluctuating" form \(\exists !\text{Nc}ʻ\alpha\) as though it possessed a "permanent" truth-value.

A fluctuating form however often possesses what is here termed a "stable" truth-value, which is as important as the permanent truth-value of other forms. For example, anticipating our definitions of elementary arithmetic, consider \(2 +_{c} 3 = 5\). There is no abstract logical proof that there are two individuals; so suppose 2 and 3 refer to classes of individuals, but 5 refers to classes of a high enough type, then with these determinations \(2 +_{c} 3 = 5\) cannot be proved. But \(2 +_{c} 3 = 5\) has a stable truth-value, since it can always be proved when all the types are high enough. In this case the fact that our empirical census of individuals (at least of the "relative" individuals of ordinary life) has outrun the capacity of logical proof, makes the fluctuation in the truth-value of the form to be entirely unimportant.

In order to make this idea precise, it is necessary to have a convention as to the order in which the types of symbols in a symbolic form are assigned. The rule we adopt is that the types of the real variables are to be first assigned, and then those of the constant symbols. The types of the apparent variables, if any, will then be completely determinate.

[Pg xvi]

A symbolic form has a stable truth-value if, after any assignment of types to the real variables, types can be assigned to the constant symbols so that the truth-value of the proposition thus obtained is the same as the truth-value of any proposition obtained by modifying it by the assignment of higher types to some or all of the constant symbols. This truth-value is the stable truth-value.

II. Formal Numbers.

The conventions, which we shall give below as to the assignment of types, practically restrict our interpretation of fluctuating symbolic forms to types in which the forms possess their stable truth-value. The assumption that these truth-values are stable never enters into the reasoning. But we judge a truth-value to be stable when any method of raising the types of the constant symbols by one step leaves it unaltered.

In practice the fluctuation of truth-values only enters into our consideration through a limited number of symbols called "formal numbers."

Formal numbers may be "constant" or "functional."

A constant formal number is any constant symbol for which there is a constant \(\alpha\) such that, in whatever type the constant symbol is determined, it is, in that type, identical with \(\text{Nc}ʻ\alpha\). In other words if \(\sigma\) be a constant symbol, then \(\sigma\) is a formal number provided that "truth" is the permanent truth-value of \(\sigma =\text{Nc}ʻ\alpha\), for some constant \(\alpha\).

The functional formal numbers are defined by enumeration; they are \[ \text{Nc}ʻ\alpha,\,\, \Sigma \text{Nc}ʻ\kappa,\,\, \Pi \text{Nc}ʻ\kappa,\,\, \text{ sm }ʻʻ\mu,\,\, \mu+_{c}\nu,\,\, \mu-_{c}\nu,\,\, \mu \times_{c}\nu,\,\, \mu^\nu, \] where in each formal number the symbols \(\alpha\), \(\kappa\), \(\mu\), \(\nu\) occurring in it are called the arguments of the functional form even when they are complex symbols. The argument of \(\text{Nc}ʻ(\alpha +\beta)\) is \(\alpha +\beta\), and those of \(\mu+_{c} (\nu +_{c} \varpi)\) are \(\mu\) and \(\nu+_{c} \varpi\), and those of \(1 +_{c} 2\) are 1 and 2.

Thus among the constant formal numbers are \[ 0,\,\, 1,\,\, 2,\,\, \ldots,\,\, \aleph_{0},\,\, 1 +_{c} 2,\,\, 2 \times _{c} \aleph_{0},\,\, 2^{2}. \] The references which support this statement are

*101·11·21·32.*123·36.*110·42.*113·23.*116·23.

Among the functional formal numbers are \[ \text{Nc}ʻ(\alpha +\beta),\,\, \mu +_{c} (\nu +_{c} \varpi),\,\,(\mu +_{c} \nu) \times _{c} \varpi,\,\, (\mu +_{c} \nu )^\varpi. \] It will be observed that e.g. \(1 +_{c} 2\) is both a constant and a functional formal number, so that the two classes are not mutually exclusive. In fact they possess an indefinite number of members in common.

All the formal numbers, with the exception of \(\text{ sm }ʻʻ\mu\) and \(\mu -_{c} \nu\), are members of \(\text{NC}\) without any hypothesis [cf. *100·41·01.*110·42.*112·101.*113·23.*114·1.*116·23, note to *119·12, and *120·411].

A functional formal number consists of two parts, namely, its argument or arguments, and the constant "form." An argument of a functional formal number may be a complex symbol, and may be constant or variable. Thus \(\mu +_{c} \nu\) is an argument of \((\mu +_{c} \nu) +_{c}\rho\), [Pg xvii]and of (\(\mu +_{c} \nu) \times _{c} 1\) and of (\(\mu +_{c}\nu)^\rho\); also \(2 +_{c} 3\) is an argument of (\(2 +_{c} 3)\times_{c} 1\). The constant form is constituted by the other symbols which are constants. Two occurrences of functional formal numbers are only occurrences of the same formal number if the arguments and also the constant forms are identical in symbolism. Thus two occurrences of \(\text{Nc}ʻ\alpha\) are occurrences of the same formal number, even if they are determined to be in different types; but \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are different formal numbers. Also \(\mu^{1}\) and \(\mu \times _{c} 1\) are different formal numbers because their "forms" are different, though the arguments \(\mu\) and 1 are the same and (in the same type) the entity denoted is the same. Thus the distinction between formal numbers depends on the symbolism and not on the entity denoted, and in considering them it is symbolic analogy and not denotation which is to be taken into account. For example two different occurrences of the same formal number will not denote the same entity, if in the two occurrences the ambiguity of type is determined differently.

The functional formal numbers are divided into three sets: (i) the primary set consisting of the forms \(\text{Nc}ʻ\alpha\), \(\Sigma \text{Nc}ʻ\kappa \), \(\Pi \text{Nc}ʻ\kappa\), (ii) the argumental set consisting only of \(\text{ sm }ʻʻ\mu\), (iii) the arithmetical set consisting of \(\mu +_{c} \nu\), \(\mu \times _{c}\nu\), \(\mu^\nu\), and \(\mu -_{c} \nu\).

A functional formal number has at most two arguments. But an argument of a functional formal number may itself be a functional formal number, and will accordingly possess either one or two arguments, which in their turn may be functional formal numbers, and so on. The whole set of arguments and of arguments of arguments, thus obtained, is called the set of components of the original formal number. Thus \(\mu\), \(\nu\), \(\rho\), and \(\mu +_{c} \nu\) are components of (\(\mu +_{c} \nu) +_{c} \rho\); and \(\mu\), \(\nu\) and \(\text{ sm }ʻʻ\mu\) are components of \(\nu +_{c} \text{ sm }ʻʻ\mu\); and \(\mu\), \(\alpha\) and \(\text{Nc}ʻ\alpha\) are components of \(\mu +_{c} \text{Nc}ʻ\alpha\). The two arguments of (\(\mu +_{c} \nu) +_{c}\rho\) are \(\mu +_{c} \nu\) and \(\rho\), and those of \(\nu +_{c}\text{ sm }ʻʻ\mu\) are \(\nu\) and \(\text{ sm }ʻʻ\mu\), and those of \(\mu+_{c} \text{Nc}ʻ\alpha\) are \(\mu\) and \(\text{Nc}ʻ\alpha\).

[Pg xviii]

Addition, multiplication, exponentiation, and subtraction will be called the arithmetical operations; and in \(\mu +_{c} \nu\), \(\mu \times _{c} \nu\), \(\mu ^\nu\), \(\mu -_{c} \nu\), \(\mu\) and \(\nu\) will each be said to be subjected to these respective operations. The arithmetical components of an arithmetical formal number (i.e. one belonging to the arithmetical set) consist of those of its components which do not appear in the capacity of components of a component which does not belong to the arithmetical set. Thus \(\mu\), \(\nu\), \(\rho\), \(\mu +_{c} \nu\) are arithmetical components of \((\mu +_{c} \nu) +_{c} \rho\); and \(\nu\) and \(\text{ sm }ʻʻ\mu\) are arithmetical components of \(\nu +_{c} \text{ sm }ʻʻ\mu\), but \(\mu\) is not one; and \(\mu\) and \(\text{Nc}ʻ\alpha\) are arithmetical components of \(\mu +_{c} \text{Nc}ʻ\alpha\), but \(\alpha\) is not one; and \(\mu\) and \(\text{ sm }ʻʻ(\nu +_{c} \rho)\) are arithmetical components of \(\mu +_{c} \text{ sm }ʻʻ(\nu +_{c} \rho)\), but \(\nu +_{c} \rho\) and \(\nu\) and \(\rho\) are components of \(\text{ sm }ʻʻ(\nu +_{c} \rho)\) and are therefore not arithmetical components of \(\mu +_{c} \text{ sm }ʻʻ(\nu +_{c} \rho)\). Only arithmetical formal numbers possess arithmetical components.

A formal number of the arithmetical set having no components which are formal numbers of the argumental set is called a pure arithmetical formal number. For example \(\mu +_{c}(\nu +_{c} \rho)\) and \(\mu +_{c} \text{Nc}ʻ\alpha\) are pure, but \(\mu +_{c} \text{ sm }ʻʻ(\nu +_{c} \rho)\) and \(\mu +_{c}\text{ sm }ʻʻ\text{Nc}ʻ\alpha\) are not pure.

There are many types involved in the consideration of a formal number. For example, in \(\text{Nc}ʻ\alpha\) there is the type of \(\text{Nc}ʻ\alpha\) and of \(\alpha\); in \(\mu +_{c} \nu\) there is the type of \(\mu +_{c} \nu\), the type of \(\mu\), and the type of \(\nu\); and so on for more complex formal numbers. The type of a formal number as a whole in any occurrence is called its actual type. This is the type of the entity which it then represents.

The other types involved in a formal number in any occurrence are called its subordinate types.

The actual types are not indicated in the symbolism for the various formal numbers as stated above. They can be indicated relatively to the type of the variable \(\xi\) by writing \(\text{Nc}(\xi)ʻ\alpha\), \(\text{ sm }_{\xi }ʻʻ\mu\), (\(\mu \times _{c} \nu)_{\xi}\), (\(\mu^\nu)_{\xi}\), (\(\mu -_{c} \nu)_{\xi}\), by the notation of *65. Even when the actual type of a complex formal number, such as \(\mu +_{c}(\nu +_{c} \varpi)\), is settled—so for instance that we have \(\{\mu +_{c} (\nu +_{c} \varpi)\}_{\xi}\)—the meaning of the symbol is not completely determined, for the type of \(\nu +_{c} \varpi\) remains ambiguous. It follows, however, from

*100·511.*110·23.*113·26.·62,

that the subordinate types make no difference to the value of a formal number, so long as the components are not null.

We can therefore make a formal number definite as soon as its actual type is definite by securing that its components are not null. This is done by the convention II T (below) combined with the definitions

*110·03·04.*113·04·05.*116·03·04.

When the subordinate types are adjusted in accordance with these definitions and conventions, they will be said to be normally adjusted.

[Pg xix]

But in order to state this convention \(\text{IIT}\) we require a definition of what is here called the adequacy of the actual type of a formal number. The general idea of adequacy is simple enough, namely that, given the subordinate types of \(\sigma\), the actual type of \(\sigma\) should be high enough to enable us logically to prove \(\exists!\sigma\) when such a proof is possible for types which are not too low. For example, all types except the lowest for which it has meaning are adequate for the constant formal number 2. It is rather difficult however to state the meaning of adequacy with precision in a manner adapted to all formal numbers. Fortunately the definition of the lowest type which corresponds to this general idea of adequacy is not important for our purposes. It will be sufficient to define as adequate some types which certainly do have the property in question.

The method of definition which we adopt is to replace the formal number \(\sigma\) by another one \(\sigma'\) so related to \(\sigma\) that with the same actual type for both we can prove \(\exists!\sigma'.\supset.\exists!\sigma\), whenever \(\sigma\) is not equal to \(\Lambda\) in all types. If \(\sigma\) be functional, we need only consider its argument, or its two arguments, and can dismiss from consideration the other components; then we replace these arguments by others so that the \(\sigma'\) has the required property. Thus:

(i) The actual types of \(\text{Nc}ʻ\alpha\), \(\Sigma\text{Nc}ʻ\kappa\), \(\Pi \text{Nc}ʻ\kappa\), and \(\text{ sm }ʻʻ\mu\) are adequate when we can logically prove \[ \exists !Ncʻt_{0}ʻ\alpha,\,\, \exists !\Sigma Ncʻt_{0}ʻ\kappa,\,\, \exists !\Pi Ncʻt_{0}ʻ\kappa,\,\, \text{and}\,\, \exists !\text{ sm }ʻʻt_{0}ʻ\mu; \]

(ii) The actual types of \(\mu +_{c} \nu\), \(\mu -_{c} \nu\), \(\mu\times _{c} \nu\), and \(\mu^\nu\) are adequate when we can logically prove \[ \begin{aligned} &\exists!\text{N}_{0}cʻt_{1}ʻ\mu +_{c} \text{N}_{0}cʻt_{1}ʻ\nu,\quad \exists !\text{N}_{0}cʻt_{1}ʻ\mu -_{c} 0\cap t_{0}ʻ\nu,\\ &\exists!\text{N}_{0}cʻt_{1}ʻ\mu \times _{c} \text{N}_{0}cʻt_{1}ʻ\nu,\quad \text{and}\quad \exists!\text{N}_{0}cʻt_{1}ʻ\mu^{{\text{N}_{0}cʻt_{1}ʻ\nu}}. \end{aligned} \]

It will be noticed that \(t_{0}ʻ\alpha\), \(t_{0}ʻ\kappa\), and \(t_0ʻ\mu\) are the greatest classes of the same type as \(\alpha\), \(\kappa\), and \(\mu\) respectively, and that \(\text{N}_{0}cʻt_{1}ʻ\mu\) and \(\text{N}_0cʻt_1ʻ\nu\) are the greatest cardinal numbers of the same type as \(\mu\) and \(\nu\) respectively. These definitions hold even when any of \(\alpha\), \(\kappa\), \(\mu\), \(\nu\) are complex symbols.

The remaining formal numbers which are not functional must certainly be constant. The difficulty which arises here is that if \(\sigma\) be such a formal number and \(\aleph_{0}\) occurs in its symbolism, we have no logical method of deciding as to the truth or falsehood of \(\exists !\aleph_{0}\) in any type. But we replace \(\aleph_{0}\) by \(\text{N}_{0}cʻt_{1}ʻ\aleph_{0}\) which is the greatest existent cardinal of the same type as \(\aleph_{0}\) in that occurrence. Thus:

(iii) If \(\sigma\) be a formal number which is not functional, an adequate actual type of \(\sigma\) is one for which we can logically prove \(\exists !\sigma'\), where \(\sigma'\) is derived from \(\sigma\) by replacing any occurrence of \(\aleph_{0}\) in \(\sigma\) by \(\text{N}_{0}cʻt_{1}ʻ\aleph_{0}\). Accordingly if \(\aleph_{0}\) does not occur in \(\sigma\), an adequate type is any actual type for which we can logically prove \(\exists !\sigma\).

In the case of members of the primary and argumental groups we have substituted the \(\text{V}\) of the appropriate type in the place of each variable. When the actual type is adequate we have \[ (\alpha ).\exists !\text{Nc}ʻ\alpha,\quad (\kappa ).\exists !\Sigma \text{Nc}ʻ\kappa,\quad (\kappa ).\exists !\Pi \text{Nc}ʻ\kappa,\quad (\mu).\exists !\text{ sm }ʻʻ\mu. \]

In the case of members of the arithmetical group (except in the case of \(\mu -_c \nu)\), we have substituted for each argument the largest cardinal number which can be obtained in the type of that argument, namely the \(\text{N}_{0}cʻ\text{V}\) for the \(\text{V}\) of the appropriate type. Accordingly we are sure (except in the case of \(\mu -_c \nu)\) that for all other values of the arguments which are existent cardinal numbers the formal number is not null.

It will be noticed that normal adjustment only concerns the subordinate types. For example *110·03 secures that in \(\text{Nc}ʻ\alpha +_{c}\mu\) the actual type of[Pg xx] \(\text{Nc}ʻ\alpha\) is adequate, and *110·23 shows that any adequate actual type of \(\text{Nc}ʻ\alpha\) will do. But nothing is said about the actual type of \(\text{Nc}ʻ\alpha +_{c}\mu\). We make the following definition: When the subordinate types of a formal number are normally adjusted, and the actual type is adequate, the types of the formal number are said to be arithmetically adjusted.

We notice that for the primary set, the arithmetical adjustment of types means the same thing as the adequate adjustment of the actual type. Also if the arguments of a formal number of the arithmetical set are simple symbols, the two ideas come to the same thing.

In the case of variable formal numbers of the primary set, it follows from *117·22·32 that when their types are arithmetically adjusted they are not equal to \(\Lambda\) for any values of their variables.

Also in the case of those variable formal numbers which are of the pure arithmetical set (excluding \(\mu -_{c} \nu\)) it follows from *100·4·52·42.*113·23.*116·23 that, working from the ultimate components reached by successive analysis upwards, for all values of such ultimate components which are members of \(\text{NC} - \iotaʻ\Lambda\) they can be reduced to the case of the formal numbers of the primary group; and that therefore they are not equal to \(\Lambda\) when their types are arithmetically adjusted. For example in \(\mu +_{c} \{\nu+_{c} (\rho +_{c} \sigma)\}\), \(\mu\), \(\nu\), \(\rho\), \(\sigma\) are these ultimate components; let them be existent cardinal numbers. Hence when the types are arithmetically adjusted, the actual type of \(\rho +_{c} \sigma\) is adequate and \(\rho +_{c} \sigma\) is an existent cardinal; we can therefore substitute \(\text{N}_{0}cʻ\alpha\) for it. By the same reasoning we can substitute \(\text{N}_{0}cʻ\beta\) for \(\nu +_c\text{N}_0cʻ\alpha\), and again \(\text{N}_{0}\text{c}ʻ\gamma\) for \(\mu +_{c} \text{N}_{0}cʻ\beta\).

A definite standard arithmetical adjustment of types for any formal number can always be found by making every use of \(\text{ sm }\), whether explicit or concealed in \(\text{Nc}\) or in some other symbol, to be homogeneous. Proofs which apply to any arithmetical adjustment of types start by dealing with this standard type, and then by the use of *104·21.*106·21·211·212·213 the extension is made to the adjacent higher classical and relational types. We then "see" that by the analogy of symbolism this extension can always be formally proved at each stage, so that we are dealing with the stable truth-value. For some constant formal numbers a lower existential type can be found than that indicated by this method.

III. Classification of Occurrences of Formal Numbers.

A symbolic form of any of the kinds [cf. *117·01·04·05·06] \[ \mu > \nu,\quad \mu < \nu,\quad \mu \geq \nu,\quad \mu \leq \nu, \] is called an arithmetical inequality.

[Pg xxi]

These forms only arise when we are comparing cardinal numbers in respect to the relation of being "greater than" or "less than." It might seem natural to include equations among these arithmetical inequalities. Their use however, even as between cardinal numbers, is not so exclusively arithmetical, and it is convenient to consider them separately under another heading during our preliminary investigations.

In the arithmetical inequalities as above written, \(\mu\) and \(\nu\), or any symbols replacing \(\mu\) and \(\nu\), are called the opposed sides of the inequality, and either of \(\mu\) or \(\nu\) is called a side of the inequality.

Symbolic forms of the kinds \(\sigma =\kappa\) and \(\sigma \neq \kappa\), where either \(\sigma\) or \(\kappa\) is a formal number, will be called equations and inequations respectively; and \(\sigma\) and \(\kappa\) are called the opposed sides of the equation or inequation, and either of them is simply a side of the equation or inequation.

When we reach the exclusively arithmetical point of view, it will be convenient to put together equations, inequations and arithmetical inequalities as one sort of symbolic form. Their separation here is for the sake of investigations into the exceptions due to the failure of existence theorems in low types. It is unnecessary to consider arithmetical inequalities in this connection.

The ways in which a symbol \(\sigma\) can occur in a symbolic form are named as follows:

The occurrence of \(\sigma\) in \(\text{ sm }ʻʻ\sigma\) is called an argumental occurrence,

The occurrence of \(\sigma\) as an argument of an arithmetical formal number (which may be a component of another formal number) or as one side of an arithmetical inequality is called an arithmetical occurrence,

The occurrence of \(\sigma\) as one side of an equation is called an equational occurrence,

The occurrence of \(\sigma\) in "\(\xi \in \sigma\)" is called an attributive occurrence,

Any other occurrence of \(\sigma\) is called a logical occurrence, so also is \(\sigma =\Lambda\).

It is obvious that a pair of opposed sides of an equation or inequation must be of the same type. Furthermore, if \(\sigma\) be a formal number, and *20·18 is applied so as to give \[ \vdash \colon\ldotp \sigma =\kappa .\supset :f(\sigma).\equiv .f(\kappa), \] [Pg xxii]the equational occurrence of \(\sigma\) must be of the same type as its occurrence in \(f(\sigma\)), otherwise the inference is fallacious. Accordingly substitution in arithmetical formulae can only be undertaken when the conventions as to the relations of ambiguous types secure this identity. This question is considered later in this prefatory statement, and the result appears in the text as *118·01.

At this point some examples will be useful; they will also be referred to subsequently in connection with the conventions limiting ambiguities of type.

*100·35. \[\begin{aligned}\vdash \colon\ldotp \exists !\text{Nc}ʻ\alpha .\lor.&\exists !\text{Nc}ʻ\beta :\supset:\\ &\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\equiv .\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta\end{aligned}\]

Here the formal numbers are \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\), each of which has three occurrences. The first occurrence of \(\text{Nc}ʻ\alpha\) is logical, its second is equational, and its third is attributive.

*100·42 (in the demonstration).

\(\vdash :\mu ,\nu \in \text{NC}.\exists !\mu \cap \nu .\supset .(\exists \alpha,\beta).\mu =\text{Nc}ʻ\alpha .\nu =\text{Nc}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \)

Here \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are the only formal numbers, and all their occurrences are equational.

*100·44 (in the demonstration).
\[ \vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .\supset .(\exists \beta ).\mu =\text{Nc}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \]

Here \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are the only formal numbers; the first occurrence of \(\text{Nc}ʻ\alpha\) is logical, its second is equational; both the occurrences of \(\text{Nc}ʻ\beta\) are equational.

*100·511. \(\vdash :\exists !\text{Nc}ʻ\beta .\supset .\text{ sm }ʻʻ\text{Nc}ʻ\beta =\text{Nc}ʻ\beta\)

Here the formal numbers are \(\text{Nc}ʻ\beta\) and \(\text{ sm }ʻʻ\text{Nc}ʻ\beta\). The first occurrence of \(\text{Nc}ʻ\beta\) is logical, the second is argumental, the third is equational; the only occurrence of \(\text{ sm }ʻʻ\text{Nc}ʻ\beta\) is equational.

*100·521. \(\vdash :\mu \in \text{NC}.\exists !\,\text{ sm }ʻʻ\mu .\supset .\text{ sm }ʻʻ\text{ sm }ʻʻ\mu =\mu\)

Here \(\text{ sm }ʻʻ\mu\) and \(\text{ sm }ʻʻ\text{ sm }ʻʻ\mu\) are the only formal numbers; \(\text{ sm }ʻʻ\mu\) has two occurrences, the first logical, the second argumental; \(\text{ sm }ʻʻ\text{ sm }ʻʻ\mu\) has one occurrence, which is equational.

*101·28 (in the demonstration).
\[ \vdash :\gamma\in \text{ sm }ʻʻ1.\equiv .(\exists \alpha ).\alpha \in 1.\gamma\, \text{ sm }\,\alpha \]

Here the formal numbers are 1 and \(\text{ sm }ʻʻ1\). The first occurrence of 1 is argumental, the second is attributive; the occurrence of \(\text{ sm }ʻʻ1\) is attributive.

*101·38. \(\vdash :\exists !2.\supset .sʻ\text{Cl}ʻʻ2=0\cup 1\cup 2\)

Here the formal numbers are 0, 1, and 2, and their occurrences are all logical.

*110·54. \(\vdash .(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{0} \text{Nc}ʻ\gamma = \text{Nc}ʻ(\alpha +\beta +\gamma)\)

Here the formal numbers are \[ \text{Nc}ʻ\alpha,\,\, \text{Nc}ʻ\beta,\,\, \text{Nc}ʻ\gamma,\,\, \text{Nc}ʻ(\alpha +\beta +\gamma),\,\, \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta,\,\, (\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{c} \text{Nc}ʻ\gamma. \] The occurrence of \(\text{Nc}ʻ(\alpha +\beta +\gamma\)) and that of (\(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{c} \text{Nc}ʻ\gamma\) are both equational, and they must be of the same type since they are opposed sides of the same equation. The occurrences of the other formal numbers[Pg xxiii] are as arithmetical components of a more complex arithmetical formal number and are therefore arithmetical.

*116·63. \(\vdash .\mu^{\nu \times _{c} \varpi} = (\mu^{\nu})^{\varpi}\)

The formal numbers are \(\nu \times _{c} \varpi\), \(\mu^{\nu }\), \(\mu ^{\nu \times _{c} \varpi}\), and (\(\mu^{\nu})^{\varpi}\). Each formal number occurs once only. The occurrences of \(\nu \times _{c} \varpi\) and \(\mu^{\nu}\) are arithmetical, and those of the other two are equational.

*117·108. \(\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \equiv : \text{Nc}ʻ\alpha > \text{Nc}ʻ\beta . \lor . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta\)

The formal numbers are \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\), each with three occurrences. The first two occurrences of each formal number are arithmetical, the last occurrence of each is equational.

*120·53 (in the demonstration).
\[ \vdash : \beta = \gamma +_{c} \delta . \exists ! \beta . \supset . \alpha ^\beta = \alpha^\gamma \times _{c} \alpha^\delta \]

Here the formal numbers are \(\gamma +_{c} \delta\), \(\alpha ^\beta\), \(\alpha ^\gamma\), \(\alpha ^\delta\), \(\alpha ^\gamma \times _{c}\alpha ^\delta\). Each formal number has one occurrence. Those of \(\gamma +_{c}\delta\), \(\alpha ^\beta\) and \(\alpha ^\gamma \times _{c} \alpha^\delta\) are equational, and those of \(\alpha ^\gamma\) and \(\alpha^\delta\) are arithmetical.

*120·53 (in the demonstration).
\[ \vdash : \alpha^\beta = \alpha^\gamma . \beta = \gamma +_{c} \delta . \exists ! \alpha^\beta . \supset . \alpha ^\gamma = \alpha ^\gamma \times _{c} \alpha ^\delta \]

Here the formal numbers are \(\alpha ^\beta\), \(\alpha ^\gamma\), \(\alpha ^\delta\), \(\alpha ^\gamma \times _{c} \alpha ^\delta\), \(\gamma +_{c} \delta\). The first occurrence of \(\alpha ^\beta\) is equational, its second occurrence is logical; the first two occurrences of \(\alpha ^\gamma\) are equational, its third occurrence is arithmetical; the only occurrence of \(\alpha ^\delta\) is arithmetical; the only occurrences of \(\alpha ^\gamma \times _{c}\alpha ^\delta\) and of \(\gamma +_{c} \delta\) are equational.

IV. The Conventions \(\text{IT}\) and \(\text{IIT}\).

Two occurrences of a formal number with the same actual type are said to be bound to each other.

The choice of types for formal numbers, when they are not made definite in terms of variables by the notation of *65, is limited by the following conventions, which enable us to dispense largely with the elaboration produced by the definition of types.

\(\text{IT}\). All logical occurrences of the same formal number are in the same type; argumental occurrences are bound to logical and attributive occurrences; and, if there are no argumental occurrences, equational occurrences are bound to logical occurrences.

This rule only applies, so far as meaning permits, to those types which remain ambiguous after the assignment of types to the real variables.

It will be noticed that if there are no argumental or logical occurrences of a formal number, \(\text{IT}\) does not in any way apply to the assignment of types to the occurrences in the form of that formal number.

[Pg xxiv]

The identification of types in argumental and attributive occurrences by \(\text{IT}\) is rendered necessary to secure the use of the equivalence \[ \gamma \in \text{ sm }ʻʻ\sigma .\equiv .(\exists \alpha ).\alpha \in \sigma .\gamma\,\text{ sm }\,\alpha, \] where \(\sigma\) is a formal number. Without the convention, this application of *37·1 would be fallacious. The only one of our examples to which this part of the convention applies is *101·28 (demonstration), where it secures that the two occurrences of 1 are in the same type. It is relevant however to the symbolism in the demonstration of *100·521.

It will be found in practice that this convention relates the types of occurrences in the same way as would naturally be done by anyone who was not thinking of the convention at all. To see how the convention works, we will run through the examples which have already been given above.

In *100·35, \(\text{IT}\) directs the logical and equational occurrences of \(\text{Nc}ʻ\alpha\) to be in the same type, and similarly for \(\text{Nc}ʻ\beta\). Also "meaning" secures that the equational types of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are the same. Thus these four occurrences are all in one type, which has no necessary relation to the types of the attributive occurrences of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\). Thus, using the notation of *65·04 to secure typical definiteness, *100·35 is to mean \[\begin{aligned} \vdash \colon\ldotp \exists !&\text{Nc}(\xi )ʻ\alpha .\lor.\exists !\text{Nc}(\xi)ʻ\beta :\supset :\\ &\text{Nc}(\xi)ʻ\alpha =\text{Nc}(\xi)ʻ\beta .\equiv .\alpha \in \text{Nc}(\alpha)ʻ\beta .\equiv .\beta \in \text{Nc}(\beta)ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta.\end{aligned}\]

The types of these attributive occurrences are settled by the necessity of "meaning."

In *100·42 (demonstration), since all the occurrences of formal numbers are equational, \(\text{IT}\) produces no limitation of types.

In *100·44 (demonstration), \(\text{IT}\) secures that the two occurrences of \(\text{Nc}ʻ\alpha\) are in the same type. Also we notice that the first occurrence of \(\text{Nc}ʻ\beta\) is really (cf. *65·04) \(\text{Nc}(\alpha)ʻ\beta\), since "\(\alpha \in \mu\)" occurs, and thus "meaning" requires this relation of types, and the second occurrence of \(\text{Nc}ʻ\beta\) is in the type of the occurrences of \(\text{Nc}ʻ\alpha\).

In *100·511, \(\text{IT}\) directs that the logical and argumental occurrences are to have the same type. In *100·521, \(\text{IT}\) directs that the two occurrences of \(\text{ sm }ʻʻ\mu\) are to have the same type. In *101·28 both occurrences of 1 are to be in the same type. In *101·38, \(\text{IT}\) directs that all the occurrences of 2 are to have the same type.

The convention \(\text{IT}\) in no way limits the types in *110·54, nor in *116·63, nor in *117·108.

In the first example from *120·53 (in the demonstration) convention \(\text{IT}\) has no application.

In the second example from *120·53 (in the demonstration) convention \(\text{IT}\) directs that the two occurrences of \(\alpha ^\beta\) shall be in the same type; and the[Pg xxv] necessity of "meaning" secures that the first occurrence of \(\alpha ^\gamma\) shall also be in this type. The same necessity secures that \(\gamma +_{c} \delta\) shall be in the same type as \(\beta\); and it also secures that in "\(\alpha ^\gamma = \alpha ^\gamma \times _{c} \alpha ^\delta\)" the first occurrence of \(\alpha ^\gamma\) and that of \(\alpha ^\gamma\times_c \alpha^\delta\) shall have a common type, which is otherwise unfettered; also nothing has been decided as to the types of \(\alpha ^\gamma\) and \(\alpha ^\delta\) in \(\alpha ^\gamma\times _{c} \alpha ^\delta\).

We now come to conventions embodying the outcome of arithmetical ideas. The term "arithmetical" is here used to denote investigations in which the interest lies in the comparison of formal numbers in respect to equality or inequality, excluding the exceptional cases—whenever the cases are exceptional—due to the failure of existence in low types. The thorough-going arithmetical point of view, which we adopt later in the investigation on Ratio and Quantity and also in this volume in *117 and *126 and some earlier propositions, would sweep aside as uninteresting all investigation of the exact ways in which the failure of existence theorems is relevant to the truth of propositions, thus concentrating attention exclusively on stable truth-values. But the logical investigation has its own intrinsic interest among the principles of the subject. It is obvious however that it should be restrained to a consideration of the theorems of purely logical interest. In practice this extrusion of uninteresting cases of the failure of arithmetical theorems, even amid the logical investigations of the first part of this volume, is effected by securing that all arithmetical occurrences of formal numbers have their actual types adequate.

As far as formal numbers of the primary group, i.e. \(\text{Nc}ʻ\alpha\), \(\Sigma \text{Nc}ʻ\kappa\), \(\Pi\text{Nc}ʻ\kappa\), are concerned, the arithmetical adjustment of types is secured formally in the symbolism by the definitions *110·03·04 for addition, and *113·04·05 for multiplication, and *116·03·04 for exponentiation, and *117·02·03 for arithmetical inequalities, and *119·02·03 for subtraction.

We save the symbolic elaboration which would arise from the extension of similar definitions to other formal numbers by the following convention:

\(\text{IIT}\). Whenever a formal number \(\sigma\) occurs, so that, if it were replaced by \(\text{Nc}ʻ\alpha\), the actual type of \(\text{Nc}ʻ\alpha\) would by definition have to be adequate, then the actual type of \(\sigma\) is also to be adequate.

For example in \(\mu +_{c} (\nu +_{c} \varpi)\), if \(\nu +_{c}\varpi\) were replaced by \(\text{Nc}ʻ\alpha\), then by *110·04 the actual type of \(\text{Nc}ʻ\alpha\) is adequate. Hence by \(\text{IIT}\) the actual type of \(\nu +_{c} \varpi\) is to be adequate: accordingly so long as \(\nu\) and \(\varpi\) are simple variables and members of \(\text{NC}-\iotaʻ\Lambda\), we can always assume \(\exists !(\nu +_{c}\varpi)\) for the type of the occurrence of \(\nu +_{c} \varpi \in \mu +_{c} (\nu +_{c} \varpi)\).

It is essential to notice that so long as the argument of an argumental formal number, or the arguments of an arithmetical formal number, are adjusted arithmetically, the exact types chosen make no difference. This follows for argumental formal numbers from *102·862·87·88, for addition from[Pg xxvi] *110·25, for multiplication from *113·26, for exponentiation from *116·26, for subtraction from *119·61·62. Thus (remembering also *100·511) in any definite type a formal number has one definite meaning provided that any subordinate formal number which occurs in its symbolism is determined existentially. The convention \(\text{IIT}\) directs us always to take this definite meaning for any pure arithmetical formal number.

The convention does not determine completely the meaning of an arithmetical formal number which is not pure. For example, \(\mu +_{c}(\nu +_{c} \rho )\) is a pure arithmetical formal number when \(\mu\), \(\nu\), \(\rho\) are determined in type; and convention \(\text{IIT}\) directs that the type of \((\nu +_{c} \rho)\) is to be adequate. But \(\mu +_{c} \text{ sm }ʻʻ(\nu +_{c} \rho)\) is an arithmetical formal number which is not pure, and convention \(\text{IIT}\) directs that the type of the domain of \(\text{ sm }\) is to be adequate, but does not affect the type of \(\nu +_{c} \rho\). Thus it is easy to see that \(\text{IIT}\) secures the adequacy of the actual types of all arithmetical components of any arithmetical formal numbers which occur, but does not affect the actual type of a formal number which occurs as the argument of an argumental formal number. But in this case convention \(\text{IT}\) will bind the actual type of this occurrence of the argument to any logical or attributive occurrence of the same formal number. For example, if \(\exists ! \nu +_{c} \rho\) and \(\mu +_c\text{ sm }ʻʻ(\nu +_c \rho)\) occur in the same form, then these two occurrences of \(\nu +_{c} \rho\) must have the same actual type. In practice argumental formal numbers are useful as components of arithmetical formal numbers for the very purpose of avoiding the automatic adjustment of types directed by \(\text{IIT}\).

The meaning of \(\text{IIT}\) is best explained by examples. Among our previous examples we need only consider those in which arithmetical formal numbers occur.

In *110·54 the convention or definitions direct us to determine the types of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) adequately when forming \(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\), also to determine \(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\) and \(\text{Nc}ʻ\gamma\) adequately when forming \((\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta)+_{c} \text{Nc}ʻ\gamma\). The convention does not apply to the types of \((\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{c} \text{Nc}ʻ\gamma\) and \(\text{Nc}ʻ(\alpha + \beta + \gamma)\). These types must be identical in order to secure meaning.

In *116·63 the convention directs us to adjust the types of \(\nu\times _c \varpi\) and \(\mu ^\nu\) adequately; it does not affect the types of \(\mu ^{\nu \times _{c} \varpi}\) and \((\mu^\nu)^\varpi\), which must be identical to secure meaning. If we replace \(\mu\), \(\nu\), \(\varpi\) by formal numbers, by 2, \(\aleph_{0}\), and 1 for example, we get "\(\vdash . 2^{\aleph_{0}\times \text{c1}} = (2^{\aleph _{0}})^{1}\)." The convention now directs that 1 is to be determined adequately. It so happens that any type is adequate for it, since \(\exists ! 1\) can be proved in any type. Then adequate types for \(\aleph _{0} \times_{c} 1\) and \(2^{\aleph _{0}}\) are types for which we can prove \(\exists! (\text{N}_{0} cʻt_{1}ʻ\aleph _{0}) \times _{c} 1\) and \(\exists! 2^{\text{N}_{0} cʻt_{1}ʻ\aleph _{0}}\). Thus if \(\tau\) is the type of \(\aleph _{0}\) in both cases, an adequate type for \(\aleph_{0} \times _{c} 1\) is \(\tau\), and for \(2^{\aleph _{0}}\) is \(\text{Cl}ʻ\tau\).

In *117·108 we find arithmetical occurrences in arithmetical inequalities. Thus \(\text{IIT}\) directs us to take the first two occurrences of \(\text{Nc}ʻ\alpha\) and the first[Pg xxvii] two of \(\text{Nc}ʻ\beta\) with adequate actual types. The type of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) in \(\text{Nc}ʻ\alpha= \text{Nc}ʻ\beta\) is not affected by it. It is evident that the conventions \(\text{IT}\), \(\text{IIT}\) are not sufficient to secure the truth of this proposition as thus symbolized. It is essential that in the equation the type be adjusted adequately for both formal numbers. In fact the general arithmetical convention, that types of equational as well as of arithmetical occurrences are adjusted arithmetically, is here used.

V. Some Important Principles.

Principle of Arithmetical Substitution. In *120·53, the application of \(\text{IIT}\) needs a consideration of the whole question of arithmetical substitution. Consider the first of the two examples. We have \[ \vdash : \beta = \gamma +_c \delta . \exists ! \beta . \supset . \alpha ^\beta = \alpha ^\gamma \times _c \alpha ^\delta. \]

It is obvious that unless we can pass with practical immediateness from "\(\beta = \gamma +_c \delta. \alpha ^\beta = \alpha ^\beta\)" to "\(\alpha ^\beta = \alpha^{\gamma +_c \delta}\)" by *20·18, arithmetic is made practically impossible by the theory of types. But a difficulty arises from the application of \(\text{IIT}\). Suppose we assign the types of our real variables first. Then the types of \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) can be arbitrarily assigned, and there is no necessary connection between them which arises from the preservation of meaning. Thus \(\beta\) may be in a type which is not an adequate type for \(\gamma +_c \delta\). Assume that this is the case. But the equational use of \(\gamma +_c \delta\) is in the same type as \(\beta\), and by \(\text{IIT}\) the arithmetical use of \(\gamma +_c \delta\) in \(\alpha^{\gamma +_c \delta}\) is in an adequate type. Thus, on the face of it, the reasoning, appealing to *20·18, by which the substitution was justified, is fallacious; for the two occurrences of \(\gamma +_c \delta\) in fact mean different things.

In order to generalize our solution of this difficulty it is convenient to define the term "arithmetical equation." An arithmetical equation is an equation between purely arithmetical formal numbers whose actual types are both determined adequately. Then it is evident that from "\(\sigma = \tau.f(\tau)\)," where \(\sigma\) and \(\tau\) are formal numbers and \(\tau\) occurs arithmetically in \(f(\tau)\), we cannot infer \(f(\sigma)\) unless the equation \(\sigma = \tau\) is arithmetical. For otherwise the \(\tau\) in the equation cannot be identified with the \(\tau\) in \(f(\tau)\).

When we have "\(\beta = \tau . f(\tau)\)," where \(\tau\) is a formal number and \(\beta\) is a number in a definite type, and wish to pass to "\(f(\beta)\)," or "\(\beta = \tau . f(\beta)\)" and wish to pass to "\(f(\tau)\)," the occurrence of \(\tau\) in \(f(\tau)\) being arithmetical, the type of \(\beta\) may not be an adequate type for \(\tau\). Accordingly the \(\tau\) in "\(\beta = \tau\)" cannot be identified with the \(\tau\) in \(f(\tau)\). The type of the \(\tau\) in the equation ought to be freed from dependence on that of \(\beta\). Accordingly the transition is only legitimate when we can write instead \[ \unicode{x201c}\beta +_c 0 = \tau . f(\tau )\unicode{x201d}\quad \text{or}\quad \unicode{x201c}\beta +_c 0 = \tau . f(\beta ),\unicode{x201d} \][Pg xxviii] where in both cases the equation is arithmetical. For now all the symbols are subject to the same rules.

If this modification can be made without altering the truth-value of the asserted propositions, the substitution is legitimate, otherwise it is not.

It is obvious that in the above our immediate passage is to or from \(f(\beta +_{c}0)\). But is easy to see that, the occurrence of \(\beta+_{c}0\) being arithmetical, we always have \[ f(\beta ).\equiv .f(\beta +_{c}0). \] In order to prove this, we have only to prove \[ \begin{array}{l} \alpha +_{c}(\beta +_{c}0)&=\alpha +_{c}\beta,\\ \alpha \times _{c}(\beta +_{c}0)&=\alpha \times _{c}\beta,\\ (\alpha +_{c}0)^\beta &=\alpha ^\beta, \alpha ^{\beta +_{c}0}&=\alpha ^\beta, \end{array} \] \[ \text{and}\qquad \alpha >\beta +_{c}0.\equiv .\alpha >\beta .\equiv .\alpha +_{c}0>\beta. \]

The demonstration of the first of these propositions runs as follows: \[ \begin{array}{l} \vdash .*110·4.\supset \vdash \colon\ldotp \beta {\sim}\in \text{NC}.\lor.\beta =\Lambda :&\supset .\beta +_{c}0=\Lambda .\alpha +_{c}\beta =\Lambda.\\ [*110·4] &\supset .\alpha +_{c}(\beta +_{c}0)=\Lambda =\alpha +_{c}\beta &\qquad \text{(1)}\\ \vdash .*110·4.&\supset \vdash \colon\ldotp \alpha {\sim}\in \text{NC}.\lor.\alpha =\Lambda :\supset .\alpha +_{c}(\beta +_{c}0)=\Lambda =\alpha +_{c}\beta &\qquad \text{(2)}\\ \vdash .*110·6.\supset \vdash :\alpha ,\beta \in \text{NC}-{℩}ʻ\Lambda .\supset .\alpha +_{c}(\beta +_{c}0)&=\alpha +_{c}\text{ sm }ʻʻ\beta \\ &=\alpha +_{c}\beta &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash :\alpha +_{c}(\beta +_{c}0)=\alpha +_{c}\beta \end{array} \]

In the above demonstration the step to (3) is legitimate since by the hypothesis \(\beta\) is a determination of \(\text{ sm }ʻʻ\beta\) in an adequate type.

Similar proofs hold for the other propositions, using *113·204 and *116·204 and *117·12 and *103·13.

We must also consider the circumstances under which we can pass from "\(\beta =\tau\)" to "\(\beta +_{c}0=\tau\)," where the latter equation is arithmetical. In other words, using *65·01 we require the hypothesis necessary for \[ \exists !\tau _{\eta }.\beta =\tau _{\xi }.\supset .\beta +_{c}0=\tau _{\eta }. \]

We have \[ \begin{array}{l} \vdash .*20·18. &\supset \vdash :\beta =\tau_{\xi }.\supset .\beta +_{c}0=\tau_{\xi }+_{c}0 &\qquad \text{(1)}\\ \vdash .*110·35. &\supset \vdash :\exists !\tau_{\xi }.\exists !\tau_{\eta }.\supset .\tau_{\xi }+_{c}0=\tau_{\eta }+_{c}0 &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash \colon\ldotp \exists !\tau_{\xi }.\exists !\tau_{\eta }.\supset :\beta =\tau_{\xi }.\supset .\beta +_{c}0=\tau_{\eta }+_{c}0 &\qquad \text{(3)}\\ \vdash .(3). &\supset \vdash \colon\ldotp \exists !\beta .\exists !\tau_{\eta }.\supset :\beta =\tau_{\xi }.\supset .\beta +_{c}0=\tau_{\eta }+_{c}0 &\qquad \text{(4)} \end{array} \]

Now in (4) the occurrences of \(\beta +_{c}0\) and \(\tau_{\eta}+_{c}0\), which are in the same type, may be chosen to be in any type we like. Hence we deduce \[ \begin{array}{l} \vdash .(4).*110·6.\supset \vdash \colon\ldotp \exists !\beta .\exists !\tau _{\eta}.\supset :\beta =\tau _{\xi}.&\supset .(\beta +_{c}0)_{\zeta }=\text{ sm }_{\zeta}ʻʻ\tau _{\eta}.\\ [*100·511] &\supset .(\beta +_{c}0)_{\zeta}=\tau_{\zeta} \end{array} \]

[Pg xxix]

Hence \(\exists !\beta\) is the requisite condition. Now since \(\zeta\) can be in any type, we can also choose it in any existential type for \(\tau\). Thus with \(\text{IIT}\) applying to the arithmetical occurrence of \(\tau\) in \(f(\tau)\), we have, where \(\tau\) is a formal number and \(\beta\) is a number in a definite type, \[ \begin{aligned} \vdash :\exists !\beta .\beta &=\tau .f(\tau).\supset .f(\beta),\\ \vdash :\exists !\beta .\beta &=\tau .f(\beta).\supset .f(\tau),\\ \vdash :\exists !\sigma .\sigma &=\tau .f(\tau).\supset .f(\sigma). \end{aligned} \]

In the last proposition by \(\text{IT}\) the equation \(\sigma =\tau\) is arithmetical. These equations are summed up in *118·01.

These three fundamental theorems embody the principle of arithmetical substitution. The hypothesis \(\exists !\beta\) is really less than is assumed in ordinary life, the usual tacit assumption being \(\beta \in\text{NC}-\iota ʻ\Lambda\). In fact unless \(\beta \in \text{NC}\), \(\beta =\tau\) is necessarily false.

Principle of Identification of Types. Suppose we have proved "\(\vdash :\text{Hp}.\supset .\phi \sigma\)" and "\(\vdash :\phi(\sigma _{\xi}).\supset .p\)," where \(\sigma\) is a formal number whose occurrence in "\(\vdash :\text{Hp}.\supset .\phi \sigma\)" is in an entirely ambiguous type, and \(\phi _{\xi}\) is the same formal number \(\sigma\) with its type related to that of \(\xi\) by *65·01. Then since the type of the \(\sigma\) in "\(\vdash :\text{Hp}.\supset.\phi \sigma\)" is ambiguous, we can write "\(\vdash :\text{Hp}.\supset.\phi (\sigma _{\xi})\)" and thence infer "\(\vdash .p\)."

The principle is: An entirely undetermined type in an asserted symbolic form can be identified with any type ambiguous or otherwise in any other asserted symbolic form or in the same symbolic form.

For example in *100·42 (demonstration) considered above, since \(\exists !\mu \cap \nu\) occurs, the first occurrences of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are of the same type, and so are their second occurrences in \(\text{Nc}ʻ\alpha=\text{Nc}ʻ\beta\). But the two types are not determined by our conventions to have any necessary connection. In fact the type in \(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\) is entirely arbitrary. Accordingly it can be identified with the other type, and thus the inference to the next line, viz. to "\(\vdash :\text{Hp}.\supset.\mu =\nu\)," is justified.

In the case of arithmetical equations, it is important to notice that we have \[ \vdash .*100·321·33.\supset \vdash \colon\ldotp \exists !\text{Nc}(\xi)ʻ\alpha .\supset :\text{Nc}(\xi)ʻ\alpha =\text{Nc}(\xi)ʻ\beta .\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta. \] Hence if \(\sigma\) and \(\tau\) are formal numbers, \[ \vdash \colon\ldotp \exists !\sigma_{\xi}.\supset :\sigma _{\xi}=\tau _{\xi}.\supset .\sigma =\tau. \] Thus if we have "\(\vdash :\text{Hp}.\exists !\sigma .\supset .\sigma=\tau\)" and "\(\vdash :\text{Hp}ʻ.\sigma _{\eta}=\tau _{\eta}.\supset.p\)," we can infer from the former proposition "\(\vdash:\text{Hp}.\exists !\sigma .\supset .\sigma _{\eta}=\tau_{\eta}\)" and from this and the latter proposition, we infer "\(\vdash:\text{Hp}'.\text{Hp}.\exists !\sigma .\supset .p\)," so the general principle of identification can be employed when the \(\phi (\sigma)\) in the first proposition is an arithmetical equation.

For example, in an example given above, *100·44 (demonstration), viz. \[ \vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .\supset .(\exists \beta).\mu =\text{Nc}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta, \][Pg xxx] the equation \(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\) is arithmetical. Accordingly we are justified in asserting the propositional function \[ \vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .\supset .(\exists \beta).\mu =\text{Nc}(\alpha )ʻ\beta .\text{Nc}(\alpha)ʻ\alpha =\text{Nc}(\alpha)ʻ\beta, \] where \(\text{Nc}(\alpha )ʻ\beta\) in "\(\mu =\text{Nc}(\alpha)ʻ\beta\)" has all along been presupposed by the necessity of meaning. Thus the inference follows, \[\begin{aligned} \vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .&\supset .\text{Nc}(\alpha )ʻ\alpha =\mu .\\ &\supset .\text{Nc}ʻ\alpha =\mu . \end{aligned}\]

This proof loses its point when \(\mu\) is looked on as a variable with necessarily the same type throughout. For then the proposition collapses into \[ \vdash \colon\ldotp \mu \in \text{NC}.\supset :\alpha \in \mu .\equiv .\text{Nc}(\alpha )ʻ\alpha =\mu . \]

But if \(\mu\) be a formal number necessarily a member of \(\text{NC}\), the proposition is really \[ \vdash \colon\ldotp \exists !\text{Nc}ʻ\alpha .\supset :\alpha \in \mu .\equiv .\text{Nc}ʻ\alpha =\mu . \]

With this presupposition we should have in the first line of the demonstration \[ \unicode{x201c}\vdash :\exists !\text{Nc}ʻ\alpha .\text{Nc}ʻ\alpha =\mu .\supset .\alpha \in \mu,\unicode{x201d} \] though with "\(\mu\)" a single variable, the line is formally correct as it stands in the text.

Recognition of Particular Cases. It is important to notice the conditions under which \(\phi \sigma\) can be recognized as a particular case of \(\phi \xi\), where \(\xi\) is a real variable and \(\sigma\) is a formal number. In the first place obviously we must substitute \(\sigma \cap t_{0}ʻ\xi\) for \(\sigma\), wherever it occurs in \(\phi \sigma\), and thus obtain \(\phi (\sigma \cap t_{0}ʻ\xi)\). Then we may find that by the application of our conventions, we can replace this by \(\phi \sigma\). For example we have

*100·42. \(\vdash :\mu ,\nu \in \text{NC}.\exists !\mu \cap \nu .\supset .\mu =\nu \)

Now put \(\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu\) for \(\mu\), we obtain \[ \begin{array}{l} \vdash :\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu ,\nu \in \text{NC}.\exists !(\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu )\cap \nu .\supset .\text{Nc}ʻ\alpha \cap t_{0}\mu =\nu &\qquad \text{(1)}\\ \vdash .(1).*100·41.\supset \vdash :\nu \in \text{NC}.\exists !\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu \cap \nu .\supset .\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu =\nu &\qquad \text{(2)} \end{array} \]

Now by \(\text{IT}\), even when \(\nu\) is a formal number, the identity of types of the two occurrences of \(\text{Nc}ʻ\alpha\) is equally secured in \[ \vdash :\nu \in \text{NC}.\exists !\text{Nc}ʻ\alpha \cap \nu .\supset .\text{Nc}ʻ\alpha =\nu. \]

Thus this is a particular case of *100·42. Such deductions can be made in general without any explicit formal statement.

[Pg xxxi]

Ambiguity of \(\text{NC}\). It follows (cf. *100·02 and *103·02) from the typical ambiguity of \(\text{Nc}\) that \(\text{NC}\) is also typically ambiguous. Hence "\(\mu \in \text{NC}.\nu \in \text{NC}\)" according to our methods of interpretation would not necessitate that \(\mu\) and \(\nu\) should be of the same type. We shall always interpret "\(\mu ,\nu \in \text{NC}\)" as standing for "\(\mu \in \text{NC}.\nu \in \text{NC}\)" and therefore as not necessarily identifying the types of \(\mu\) and \(\nu\). Similarly for \(\text{N}_{0}\text{C}\), \(\text{NC induct}\), and \(\text{NC ind}\). For example

*110·402. \(\vdash :\mu,\nu \in \text{N}_{0}\text{C}.\supset .\exists !(\mu +_{c} \nu )\cap tʻtʻ(\mu \uparrow \nu)\)

Here the \(\mu\) and \(\nu\) need not be of the same type. Again

*110·41. \(\vdash :\mu,\nu \in \text{N}_{0}\text{C}.tʻ\mu = tʻ\nu .\supset .\exists !(\mu +_{c} \nu)\cap tʻ\mu\)

Here the identification of the types of \(\mu\) and \(\nu\) requires the hypothesis "\(tʻ\mu = tʻ\nu\)."

VI. Conventions \(\text{AT}\) and \(\text{Infin T}\).

General Arithmetical Convention. Conventions \(\text{IT}\) and \(\text{IIT}\) are always applied, but the following convention is not used at first. This convention limits the remaining ambiguity of type by sweeping away the exceptional cases in low types, due to the failure of existence theorems. The convention will be cited as \(\text{AT}\).

\(\text{AT}\). All equations involving pure arithmetical formal numbers are to be arithmetical.

We have seen that from an arithmetical equation the analogous equation in any other type can be deduced. Thus with \(\text{AT}\) all equations between formal numbers are so determined in type that their truth in "any type" is deducible. Thus in the few early propositions where \(\text{AT}\) is introduced, the fact is noted by stating that the equations hold "in any type." These propositions are *103·16, *110·71·72.

The effect of applying \(\text{AT}\) to other propositions in *100 is to render some of the hypotheses (usually logical forms affirming existence) unnecessary, but also materially to limit the scope of the propositions. Take for example

*100·35. \[\begin{aligned}\vdash :\exists !\text{Nc}ʻ\alpha .\lor.&\exists !\text{Nc}ʻ\beta :\supset:\\ &\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta .\equiv .\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta\end{aligned}\]

If we apply \(\text{AT}\) to this, we can write \[ \vdash :\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta .\equiv .\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta. \]

For the equational occurrences of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are by \(\text{AT}\) and \(\text{IIT}\) to be with adequate actual types. But if \(\alpha\) is a small class in a high type, an adequate actual type for \(\text{Nc}ʻ\alpha\) will be a high type, whereas \(\exists !\text{Nc}ʻ\alpha\) may hold in a low type. Thus with \(\text{AT}\), for the sake of simplicity we abandon the statement of the minimum of hypothesis necessary for our propositions. The enunciation of no other proposition in *100 is affected.

The enunciation of no proposition in *101 is affected by \(\text{AT}\), though it would unduly limit the scope of *101·34. In *110, \(\text{AT}\) would unduly limit the scope of such propositions as

*110·22·23·24·25·251·252·3·31·32·331·34·35·351·44·51·54

and of many others, without altering their enunciations. There is no proposition in *110 whose enunciation it would alter. \(\text{AT}\) is already[Pg xxxii] applied to *110·71·72; if \(\text{AT}\) is removed from these propositions, then \(\exists !\text{Nc}ʻ\alpha\) must be added as an hypothesis to both of them. The effect of \(\text{AT}\) on *113 and *116 is entirely analogous to that on *110; in neither of these two numbers is there any proposition to which \(\text{AT}\) is applied in the text.

As regards *117, \(\text{AT}\) is applied throughout, so that the propositions are all in the form suitable for subsequent investigations in which the interest is purely arithmetical. It is important however to analyse the effect of AT on the enunciations for the sake of logical investigations, especially in connection with *120. First, \(\text{AT}\) can only affect propositions in which equations or inequations occur, and among such propositions it does not affect the enunciations of those in which both sides of the equations are not formal numbers, so that the equations are not arithmetical after the application of AT. These propositions are *117·104·14·24·241·243·31·551. These propositions, which are characterized by the presence of a single letter on one side of any equation involved, can be recognized at a glance. The propositions involving arithmetical equations whose enunciations are unaltered by the removal of \(\text{AT}\) are *117·21·54·592. Propositions involving inequations whose enunciations are unaltered by the removal of \(\text{AT}\) are *117·26·27. Finally the only propositions of *117 whose enunciations are altered by the removal of \(\text{AT}\) are *117·108·211·23·25.

In *118 and *119 \(\text{AT}\) is not used.

In *120, which is devoted to those properties of inductive cardinals which are of logical interest, \(\text{AT}\) is never used. None of the propositions *117·108·211·23·25·3 are cited in it, except *117·25 in the demonstration of *120·435 for a use where \(\text{AT}\) is not relevant. The application of AT to *120 would simplify the hypotheses of *120·31·41·451·53·55, and limit the scopes of the propositions.

One other convention, which we will call "\(\text{Infin T}\)," is required in certain propositions where the hypothesis implies that there are types in which every inductive cardinal exists, i.e. in which \(\text{V}\) is not an inductive class. Among such hypotheses are \(\text{Infin ax}\), \(\exists !\text{Prog}\), \(\exists!\aleph_{0}\) (or typically definite forms of these hypotheses), or \(R\in \text{Prog}\) or \(\alpha \in \aleph_{0}\). When such hypotheses occur, we shall assume that \(\text{NC}\) induct is, whenever significance permits, to be determined in a type in which every inductive cardinal exists, i.e. in which the axiom of infinity holds (cf. *120·03·04). The statement of this convention is as follows:

\(\text{Infin T}\). When the hypothesis of a proposition implies that there is a type in which every inductive cardinal exists, every occurrence of "\(\text{NC induct}\)" in this proposition is to be taken (if conditions of significance permit) in a sufficiently high type to insure the existence of every inductive cardinal.

It is to be observed that this convention would be unnecessary if we confined ourselves to one extensional hierarchy, for in any one such hierarchy[Pg xxxiii] all types are inductive or all are non-inductive, so that if every inductive cardinal exists in one type in the hierarchy, the same holds for any other type in the hierarchy. But when we no longer confine ourselves to one extensional hierarchy, this result may not follow. For example, it may be the case that the number of individuals is inductive, but the number of predicative functions of individuals is not inductive; at any rate, no logical reason can be given against this possibility, which can only be rejected on empirical grounds, if at all.

The way in which this convention is used may be illustrated by the demonstration of *122·33. In the second line of this demonstration, we show that the hypothesis implies \[ \begin{array}{l} \qquad\qquad\qquad\qquad\qquad \text{E}!\nu_{R} . \supset . \text{E}!(\nu +_{c}1)_{R} &\qquad \text{(1)}\\ \text{where by} *121·04\qquad \nu_{R} = \breve{R}_{\nu -_{c}1}ʻBʻR &\qquad\text{Df},\\ \text{and by} *121·02\qquad R_{\nu} = \hat{x} \hat{y}\{\text{N}_{0}\text{c}ʻR(x \vdash\dashv y) = \nu +_{c}1\} &\qquad\text{Df}. \end{array} \] It will be seen that these definitions do not suffice to determine the type of \(\nu\). Hence in (1), the \(\nu\) on the left may not be of the same type as the \(\nu +_{c}1\) on the right. Now the use of *122·473, which occurs in the next line of the demonstration of *122·33, requires that the \(\nu\) on the left and the \(\nu +_{c}1\) on the right should be of the same type. This requires that the \(\nu\) should not be taken in a type in which we have \(\exists !\nu. \nu +_c 1 = \Lambda\). Hence in order to apply *120·473, we must choose a type in which all inductive cardinals exist. Since "\(R \in \text{Prog}\)" occurs in the hypothesis, we know that all inductive cardinals exist in the type of \(CʻR\). But it is unnecessary to restrict ourselves to the type of \(CʻR\), since any other type in which all inductive cardinals exist will equally secure the validity of the demonstration. Thus the convention \(\text{Infin T}\) secures the restriction required, and no more.

The convention \(\text{Infin T}\) is often relevant when "\(\text{Infin ax}\)" without any typical determination occurs in the hypothesis. Whenever this is the case, if "\(\text{NC induct}\)" occurs in the proposition in a way which leaves its type undetermined so far as conditions of significance are concerned, it is to be taken in a type in which all its members exist.

VII. Final Working Rule in Arithmetic.

It is now (whenever \(\text{AT}\) is used, together with \(\text{Infin T}\) when necessary) possible finally to sweep aside all consideration of types in connection with inductive numbers. For by combining *126·121 *126·122 and *120·4232·4622, we see that it is always possible to take the type high enough so that no definitely determined inductive number shall be null (\(\Lambda)\), and that all the inductive reasoning can take place within this type. Furthermore we have already seen that the arithmetical operations are independent of the types of the components, so long as they are existential. Thus, as far as the ordinary[Pg xxxiv] arithmetic of finite numbers is concerned, all the conventions (including AT), and the necessity for hypotheses as to the existence of inductive numbers, are finally superseded by the following single rule:

Rule of Indefinite Numbers. The type assigned to any symbol which represents an inductive number is such that the symbol is not equal to \(\Lambda\).

We make the definition

*126·01. \(\text{Nc ind} = \text{Nc induct} - \iota ʻ\Lambda \quad \text{Df}\)

Wherever this symbol "\(\text{Nc ind}\)" for the class of "indefinite inductive cardinal numbers" is used, the above rule is adhered to. In other words, "\(\mu \in \text{Nc ind}\)" can always be replaced by "\(\mu = \text{Nc}ʻ\alpha .\alpha \in \text{Cls induct}\)," where \(\text{Nc}ʻ\alpha\) is a homogeneous or ascending cardinal, and \(\alpha\) is the appropriate constant, or is a variable, as the case may be. In the latter case, a symbolic form such as \[ (\mu).f(\mu \in \text{Nc ind},\mu) \] can be replaced by \[ (\mu,\alpha).f(\mu = \text{Nc}ʻ\alpha .\alpha \in \text{Cls induct},\mu). \]

Furthermore by *120·4622 it follows that with this rule the result of proceeding by induction in one type and then transforming to another type is the same as that of proceeding by induction in the latter type. Thus for example there is no advantage to be gained by discriminating between \(2_{\xi }\) and \(2_{\eta }\); for \(\text{ sm }_{\eta }ʻʻ2_{\xi} = 2_{\eta }\), \(\text{ sm }_{\xi }ʻʻ2_{\eta } = 2_{\xi }\), \(\mu +_{c}2_{\xi } = \mu +_{c} 2_{\eta }\), \(\mu \times _{c} 2_{\xi } = \mu \times _{c} 2_{\eta }\), \(\mu ^{2_\xi } = \mu ^{2_\eta }\), \(2_{\xi^\mu } = 2_{\eta ^\mu }\), and \(\mu \geq 2_{\xi }.\equiv .\mu \geq 2_{\eta }\), and so on.

[Pg 1]

Hence all discrimination of the types of indefinite inductive numbers may be dropped; and the types are entirely indefinite and irrelevant.

[Pg 3]

SUMMARY OF PART III.

IN this Part, we shall be concerned, first, with the definition and general logical properties of cardinal numbers (Section A); then with the operations of addition, multiplication and exponentiation, of which the definitions and formal laws do not require any restriction to finite numbers (Section B); then with the theory of finite and infinite, which is rendered somewhat complicated by the fact that there are two different senses of "finite," which cannot (so far as is known) be identified without assuming the multiplicative axiom. The theory of finite and infinite will be resumed, in connection with series, in Part V, Section E.

It is in this Part that the theory of types first becomes practically relevant. It will be found that contradictions concerning the maximum cardinal are solved by this theory. We have therefore devoted our first section in this Part (with the exception of two numbers giving the most elementary properties of cardinals in general, and of 0 and 1 and 2, respectively) to the application of types to cardinals. Every cardinal is typically ambiguous, and we confer typical definiteness by the notations of *63, *64, and *65. It is especially where existence-theorems are concerned that the theory of types is essential. The chief importance of the propositions of the present part lies, not only, as throughout the book, in the hypotheses necessary to secure the conclusions, but also in the typical ambiguity which can be allowed to the symbols consistently with the truth of the propositions in all the cases thereby included.

Summary of Section A.

The Cardinal Number of a class \(\alpha\), which we will denote by "\(\text{Nc}ʻ\alpha\)," is defined as the class of all classes similar to \(\alpha\), i.e. as \(\hat{\beta} (\beta\,\text{ sm }\,\alpha)\). This definition is due to Frege, and was first published in his Grundlagen der Arithmetik[2]; its symbolic expression and use are to be found in his Grundgesetze der Arithmetik[3]. The chief merits of this definition are (1) that the formal properties which we expect cardinal numbers to have result from it; (2) that unless we adopt this definition or some more complicated and practically equivalent definition, it is necessary to regard the cardinal number of a class as an indefinable. Hence the above definition avoids a useless indefinable with its attendant primitive propositions.

It will be observed that, if \(x\) is any object, 1 is not the cardinal number of \(x\), but that of \(\iota ʻx\). This obviates a confusion which otherwise is liable to arise in dealing with classes. Suppose we have a class \(\alpha\) consisting of many terms; we say, nevertheless, that it is one class. Thus it seems to be at once one and many. But in fact it is \(\alpha\) that is many, and \(\iotaʻ\alpha\) that is one. In regard to zero, the analogous point is still clearer. Suppose we say "there are no Kings of France." This is equivalent to "the class of Kings of France has no members," or, in our language, "the class of Kings of France is a member of the class 0." It is obvious that we cannot say "the King of France is a member of the class 0," because there is no King of France. Thus in the case of 0 and 1, as more evidently in all other cases, a cardinal number appertains to a class, not to the members of the class.

For the purposes of formal definition, we subject the formula \[ \text{Nc}ʻ\alpha = \hat{\beta} (\beta\,\text{ sm }\,\alpha) \] to some simplification. It will be seen that, according to this formula, "\(\text{Nc}\)" is a relation, namely the relation of a cardinal number to any class of which it is the number. Thus for example 1 has to \(\iota ʻx\) the relation \(\text{Nc}\); so has[Pg 5] 2 to \(\iota ʻx \cup \iota ʻy\), provided \(x\neq y\). The relation \(\text{Nc}\) is, in fact, the relation \(\overrightarrow{\text{ sm }}\); for \(\overrightarrow{\text{ sm }}ʻ\alpha = \hat{\beta} (\beta\,\text{ sm }\, \alpha)\). Hence for formal purposes of definition we put \[ \text{Nc} = \overrightarrow{\text{ sm }} \quad\text{Df}. \]

The class of cardinal numbers is the class of objects which are the cardinal numbers of something or other, i.e. of objects which, for some \(\alpha\), are equal to \(\text{Nc}ʻ\alpha\). We call the class of cardinal numbers \(\text{NC}\); thus we have \[ \text{NC} = \hat{\mu}\{(\exists \alpha) . \mu = \text{Nc}ʻ\alpha\}. \]

For purposes of formal definition, we replace this by the simpler formula \[ \text{NC} = \text{D}ʻ\text{Nc} \quad\text{Df}. \]

In the present section, we shall be concerned with what we may call the purely logical properties of cardinal numbers, namely those which do not depend upon the arithmetical operations of addition, multiplication and exponentiation, nor upon the distinction of finite and infinite[4]. The chief point to be dealt with, as regards both importance and difficulty, is the relation of a cardinal number in one type to the same or an associated cardinal number in another type. When a symbol is ambiguous as to type, we will call it typically ambiguous; when, either always or in a given context, it is unambiguous as to type, we will call it typically definite. Now the symbol "\(\text{ sm }\)" is typically ambiguous; the only limitation on its type is that its domain and converse domain must both consist of classes. When we have \(\alpha\,\text{sm}\,\beta\), \(\alpha\) and \(\beta\) need not be of the same type, in fact, in any type of classes, there are classes similar to some of the classes of any other type of classes. For example, we have \(\iota ʻx \text{ sm } \iota ʻy\), whatever types \(x\) and \(y\) may belong to. This ambiguity of "\(\text{ sm }\)" is derived from that of \(1 \rightarrow 1\), which in turn is derived from that of 1. We denote (cf. *65·01) by "\(1_{\alpha}\)" all the unit classes which are of the same type as \(\alpha\). Then (according to the definition *70·01) \(1_{\alpha} \rightarrow 1_{\beta}\) will be the class of those one-one relations whose domain is of the same type as \(\alpha\) and whose converse domain is of the same type as \(\beta\). Thus "\(1_{\alpha} \rightarrow 1_{\beta}\)" is typically definite as soon as \(\alpha\) and \(\beta\) are given. Suppose now, instead of having merely \(\gamma \text{ sm } \delta\), we have \[ (\exists R). R\in 1_{\alpha } \rightarrow 1_{\beta } . \text{D}ʻR = \gamma . \text{ᗡ}ʻR = \delta ; \] then we know not only that \(\gamma\,\text{ sm }\,\delta\), but also that \(\gamma\) belongs to the same type as \(\alpha\), and \(\delta\) belongs to the same type as \(\beta\). When the ambiguous symbol "\(\text{ sm }\)" is rendered typically definite by having its domain defined as being of the same type as \(\alpha\), and its converse domain defined as being of the same type as \(\beta\), we write it "\(\text{ sm }_{(\alpha ,\beta)}\)," because generally, in accordance with *65·1, if \(R\) is a typically ambiguous relation, we write [Pg 6]\(R_{(\alpha ,\beta)}\) for the typically definite relation that results when the domain of \(R\) is to consist of terms of the same type as \(\alpha\), and the converse domain is to consist of terms of the same type as \(\beta\). Thus we have \[ \gamma\,\text{ sm }_{(\alpha, \beta)}\,\delta . \equiv . (\exists R) . R \in 1_{\alpha} \rightarrow 1_{\beta} . \gamma = \text{D}ʻR . \delta = \text{ᗡ}ʻR. \] Here everything is typically definite if \(\alpha\) and \(\beta\) (or their types) are given.

Passing now to the relation "\(\text{Nc}\)," it will be seen that it shares the typical ambiguity of "\(\text{sm}\)." In order to render it typically definite, we must derive it from a typically definite "\(\text{sm}\)." So long as nothing is added to give typical definiteness, "\(\text{Nc}ʻ\gamma\)" will mean all the classes belonging to some one (unspecified) type and similar to \(\gamma\). If \(\alpha\) is a member of the type to which these classes are to belong, then \(\text{Nc}ʻ\gamma\) is contained in the type of \(\alpha\). For this case, it is convenient to introduce the following two notations, already defined in *65. When a typically ambiguous relation \(R\) is to be rendered typically definite as to its domain only, by deciding that every member of the domain is to be contained in the type of \(\alpha\), we write "\(R(\alpha)\)" in place of \(R\). When we further wish to determine \(R\) as having members of the converse domain contained in the type of \(\beta\), we write "\(R(\alpha, \beta)\)" in place of \(R\); and when we wish members of the converse domain to be members of the type of \(\beta\), we write "\(R(\alpha_{\beta})\)" in place of \(R\). Thus \[ \text{sg}ʻ{R_{(\alpha, \beta)}} = \{\text{sg}ʻR\}(\alpha_{\beta}) \] (cf. *65·2), and in particular, since \(\text{Nc} =\overrightarrow{\text{ sm }}\), \[ \text{Nc}(\alpha_{\beta}) = \text{sg}ʻ\text{ sm }_{(\alpha, \beta)}. \] Thus "\(\text{Nc}(\alpha_{\beta})ʻ\gamma\)" is only significant when \(\gamma\) is of the same type as \(\beta\), and then it means "classes of the same type as \(\alpha\) and similar to \(\gamma\) (which is of the same type as \(\beta\))."

"\(\text{Nc}(\alpha)ʻ\gamma\)" will mean "classes of the same type as \(\alpha\) and similar to \(\gamma\)." As soon as the types of \(\alpha\) and \(\gamma\) are known, this is a typically definite symbol, being in fact equal to \(\text{Nc}(\alpha_{\gamma})ʻ\gamma\). Hence so long as we only wish to consider "\(\text{Nc}ʻ\gamma\)," typical definiteness is secured by writing "\(\text{Nc}(\alpha)\)" in place of "\(\text{Nc}\)."

When we come to the consideration of \(\text{NC}\), "\(\text{Nc}(\alpha)\)" is no longer a sufficient determination, although it suffices to determine the type. Suppose we put \[ \text{NC}^{\beta}(\alpha) = \text{D}ʻ\text{Nc}(\alpha_{\beta}) \quad\text{Df}; \] we have also, in virtue of the definitions in *65, \[ \text{NC}(\alpha) = \text{NC} \cap t^{2}ʻ\alpha = \text{D}ʻ\text{Nc}(\alpha). \] Thus \(\text{NC}(\alpha)\) is definite as to type, but is the domain of a relation whose converse domain is ambiguous as to type; and it will appear that there are some propositions about \(\text{NC}(\alpha)\) whose truth or falsehood depends upon the determination chosen for the converse domain of \(\text{Nc}(\alpha)\). Hence if we wish to have a symbol which is completely definite, we must write [Pg 7]"\(\text{NC}^{\beta}(\alpha)\)."

This point is important in connection with the contradictions as to the maximum cardinal. The following remarks will illustrate it further.

Cantor has shown that, if \(\beta\) is any class, no class contained in \(\beta\) is similar to \(\text{Cl}ʻ\beta\). Hence in particular if \(\beta\) is a type, no class contained in \(\beta\) is similar to \(\text{Cl}ʻ\beta\), which is the next type above \(\beta\). Consequently, if \(\beta = \alpha \cup - \alpha\), where \(\alpha\) is any class, we have \[ {\sim}(\exists \gamma) . \gamma \subset \alpha \cup - \alpha . \gamma \text{ sm } \text{Cl}ʻ(\alpha \cup - \alpha). \]

Now (cf. *63) we put \[ t_{0}ʻ\alpha = \alpha \cup - \alpha \quad\text{Df}, \] and we have \(tʻ\alpha = \text{Cl}ʻ(\alpha \cup - \alpha)\). Thus we find \[ \begin{aligned} {\sim}&(\exists \gamma) . \gamma \subset t_{0}ʻ\alpha . \gamma\, \text{ sm }\, tʻ\alpha.\\ \text{Hence}\qquad &\text{Nc}(\alpha_{tʻ\alpha})ʻtʻ\alpha = \Lambda . \end{aligned} \] That is to say, no class of the same type as \(\alpha\) has as many members as \(tʻ\alpha\) has. Hence also \[ \begin{aligned} &\Lambda \in \text{NC}^{tʻ\alpha}(\alpha).\\ \text{But}\qquad \gamma \subset t_{0}ʻ\alpha . \supset . \gamma \in &\text{Nc}(\alpha_{\alpha })ʻ\gamma . \supset . \exists !\text{Nc}(\alpha_{\alpha })ʻ\gamma , \end{aligned} \] and "\(\text{Nc}(\alpha_{\alpha})ʻ\gamma\)" is only significant when \(\gamma \subset t_{0}ʻ\alpha\); hence \[ \begin{aligned} \mu \in &\text{NC}^{\alpha}(\alpha) . \supset_{\mu } . \exists !\mu\\ \text{and}\qquad\qquad &\Lambda {\sim}\in \text{NC}^{\alpha}(\alpha). \end{aligned} \]

Now the notation "\(\text{NC}(\alpha)\)" will apply with equal justice to \(\text{NC}^{\alpha }(\alpha)\) or to \(\text{NC}^{tʻ\alpha}(\alpha)\); but we have just seen that in the first case we shall have \(\Lambda {\sim}\in \text{NC}(\alpha)\), and in the second we shall have \(\Lambda \in \text{NC}(\alpha)\). Consequently "\(\text{NC}(\alpha)\)" has not sufficient definiteness to prevent practically important differences between the various determinations of which it is capable.

A converse procedure to the above yields similar results. Let \(\alpha\) be a class of classes; then \(sʻ\alpha\) is of lower type than \(\alpha\). Let us consider \(\text{NC}^{sʻ\alpha}(\alpha)\). In accordance with *63, we write \(t_{1}ʻ\alpha\) for the type containing \(sʻ\alpha\), i.e. for \(sʻ\alpha \cup - sʻ\alpha\). Then the greatest number in the class \(\text{NC}^{sʻ\alpha}(\alpha)\) will be \(\text{Nc}(\alpha)ʻt_{1}ʻ\alpha\); but neither this nor any lesser member of the class will be equal to \(\text{Nc}(\alpha)ʻt_{0}ʻ\alpha\), because, as before, \[ {\sim}(\exists \gamma ) . \gamma \subset t_{1}ʻ\alpha . \gamma \text{ sm } t_{0}ʻ\alpha . \] Hence \(\text{Nc}(\alpha )ʻt_{0}ʻ\alpha\), which is a member of \(\text{NC}^{\alpha }(\alpha)\), is not a member of \(\text{NC}^{sʻ\alpha }(\alpha)\); but \(\text{NC}^{\alpha}(\alpha)\) and \(\text{NC}^{sʻ\alpha}(\alpha)\) have an equal right to be called \(\text{NC}(\alpha)\). Hence again "\(\text{NC}(\alpha)\)" is a symbol not sufficiently definite for many of our purposes.

The solution of the paradox concerning the maximum cardinal is evident in view of what has been said. This paradox is as follows: It results from a theorem of Cantor's that there is no maximum cardinal, since, for all values of \(\alpha\), \[ \text{Nc}ʻ\text{Cl}ʻ\alpha \gt \text{Nc}ʻ\alpha . \] [Pg 8] But at first sight it would seem that the class which contains everything must be the greatest possible class, and must therefore contain the greatest possible number of terms. We have seen, however, that a class \(\alpha\) must always be contained within some one type; hence all that is proved is that there are greater classes in the next type, which is that of \(\text{Cl}ʻ\alpha\). Since there is always a next higher type, we thus have a maximum cardinal in each type, without having any absolutely maximum cardinal. The maximum cardinal in the type of \(\alpha\) is \[ \text{Nc}(\alpha)ʻ(\alpha \cup - \alpha). \] But if we take the corresponding cardinal in the next type, i.e. \[ \text{Nc}(\text{Cl}ʻ\alpha )ʻ(\alpha \cup - \alpha), \] this is not as great as \(\text{Nc}(\text{Cl}ʻ\alpha)ʻ\text{Cl}ʻ(\alpha\cup - \alpha)\), and is therefore not the maximum cardinal of its type. This gives the complete solution of the paradox.

For most purposes, what we wish to know in order to have a sufficient amount of typical definiteness is not the absolute types of \(\alpha\) and \(\beta\), as above, but merely what we may call their relative types. Thus, for example, \(\alpha\) and \(\beta\) may be of the same type; in that case, \(\text{Nc}(\alpha _{\beta})\) and \(\text{NC}^{\beta }(\alpha)\) are respectively equal to \(\text{Nc}(\alpha _{\alpha })\) and \(\text{NC}^{\alpha }(\alpha)\). We will call cardinals which, for some \(\alpha\), are members of the class \(\text{NC}^{\alpha }(\alpha)\), homogeneous cardinals, because the "\(\text{sm}\)" from which they are derived is a homogeneous relation. We shall denote the homogeneous cardinal of \(\alpha\) by "\(\text{N}_{0}\text{C}ʻ\alpha\)," and we shall denote the class of homogeneous cardinals (in an unspecified type) by "\(\text{N}_{0}\text{C}\)"; thus we put \[ \begin{aligned} \text{N}_{0}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻ\alpha &\quad\text{Df},\\ \text{N}_{0}\text{C} &= \text{D}ʻ\text{N}_{0}\text{c} &\quad\text{Df}. \end{aligned} \] Almost all the properties of \(\text{N}_{0}\text{C}\) are the same in different types. When further typical definiteness is required, it can be secured by writing \(\text{N}_{0}\text{c}(\alpha)\), \(\text{N}_{0}\text{C}(\alpha)\) in place of \(\text{N}_{0}\text{c}\), \(\text{N}_{0}\text{C}\). For although \(\text{Nc}(\alpha)\) and \(\text{NC}(\alpha)\) were not wholly definite, \(\text{N}_{0}\text{c}(\alpha)\) and \(\text{N}_{0}\text{C}(\alpha)\) are wholly definite. Apart from the fact of being of different types, the only property in which \(\text{N}_{0}\text{C}(\alpha)\) and \(\text{N}_{0}\text{C}(\beta)\) differ when \(\alpha\) and \(\beta\) are of different types is in regard to the magnitude of the cardinals belonging to them. Thus suppose the whole universe consisted (as monists aver) of a single individual. Let us call the type of this individual "\(\text{Indiv}\)." Then \(\text{N}_{0}\text{C}(\text{Indiv})\) will consist of 0 and 1, i.e. \[ \text{N}_{0}\text{C}(\text{Indiv}) = \iotaʻ0 \cup \iota ʻ1. \] But in the next higher type, there will be two members, namely \(\Lambda\) and \(\text{Indiv}\). Thus \[ \begin{aligned} &\text{N}_{0}\text{C}(tʻ\text{Indiv}) = \iotaʻ0 \cup \iotaʻ1 \cup \iotaʻ2.\\ \text{Similarly}\qquad \text{N}_{0}\text{C}&(tʻtʻ\text{Indiv}) = \iotaʻ0 \cup \iotaʻ1 \cup \iotaʻ2 \cup \iotaʻ3 \cup \iotaʻ4, \end{aligned} \][Pg 9] the members of \(tʻtʻ\text{Indiv}\) being \(\Lambda \cap tʻ\text{Indiv}, \iotaʻ\Lambda\), \(\iotaʻ\text{Indiv}\), \(\iotaʻ\Lambda \cup \iotaʻ\text{Indiv}\); and so on. (The greatest cardinal in any except the lowest type is always a power of 2.)

The maximum of \(\text{N}_{0}\text{C}(\alpha)\) is \(\text{N}_{0}\text{c}ʻt_{0}ʻ\alpha\); but apart from this difference of maximum and its consequences, \(\text{N}_{0}\text{C}(\alpha)\) and \(\text{N}_{0}\text{C}(\beta)\) do not differ in any important properties. Hence for most purposes \(\text{N}_{0}\text{C}\) and \(\text{N}_{0}\text{c}\) have as much typical definiteness as is necessary.

Among cardinals which are not homogeneous we shall consider three kinds. The first of these we shall call ascending cardinals. A cardinal \(\text{NC}^{\beta}(\alpha)\) is called an ascending cardinal if the type of \(\beta\) is \(tʻ\alpha\) or \(tʻtʻ\alpha\) or \(tʻtʻtʻ\alpha\) or etc. We write \(t^{2}ʻ\alpha\) for \(tʻtʻ\alpha\), \(t^{3}ʻ\alpha\) for \(tʻtʻtʻ\alpha\), and so on. We put \[ \begin{aligned} \text{N}^{1}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻtʻ\alpha &&\quad\text{Df}\\ \text{N}^{2}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt^{2}ʻ\alpha &&\quad\text{Df}\\ \text{N}^{3}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt^{3}ʻ\alpha &&\quad\text{Df and so on},\\ \text{and}\qquad \text{N}^{1}\text{C} &= \text{D}ʻ\text{N}^{1}\text{c} &&\quad\text{Df}\\ \text{N}^{2}\text{C} &= \text{D}ʻ\text{N}^{2}\text{c} &&\quad\text{Df}\\ \text{N}^{3}\text{C} &= \text{D}ʻ\text{N}^{3}\text{c} &&\quad\text{Df and so on}. \end{aligned} \] We then have obviously \[ \text{N}^{1}\text{C}(tʻ\alpha) \subset \text{N}_{0}\text{C}(tʻ\alpha). \] We also have (by what was said earlier) \[ \begin{aligned} &\text{N}_{0}\text{c}ʻtʻ\alpha {\sim}\in \text{N}^{1}\text{C}(tʻ\alpha).\\ \text{Hence}\qquad &\exists !\text{N}_{0}\text{C}(tʻ\alpha) - \text{N}^{1}\text{C}(tʻ\alpha). \end{aligned} \] The members of \(\text{N}_{0}\text{C}(tʻ\alpha) - \text{N}^{1}\text{C}(tʻ\alpha)\) will be all cardinals which exceed \(\text{Nc}ʻt_{0}ʻ\alpha\) but do not exceed \(\text{Nc}ʻtʻ\alpha\).

Let us recur in illustration to our previous hypothesis of the universe consisting of a single individual. Then \(\text{N}^{1}\text{c}\,\text{Indiv}\) will consist of those classes which are similar to "\(\text{Indiv}\)" but of the next higher type. These are \(\iota ʻ\Lambda\) and \(\iota ʻ\text{Indiv}\). In our case we had \(\text{N}_{0}\text{c}(\text{Indiv}) = 1\). This leads to \[ \text{N}^{1}\text{c}\,\text{Indiv} = 1 . \text{N}^{2}\text{c}\,\text{Indiv} = 1\, \quad\text{etc}. \] or, introducing typical definiteness, \[ \text{N}^{1}\text{c}\,\text{Indiv} = 1(tʻ\text{Indiv}) . \text{N}^{2}\text{c}\,\text{Indiv} = 1(t^{2}ʻ\text{Indiv})\, \quad\text{etc}. \] We have then \(1(tʻ\text{Indiv}) \in\text{N}^{1}\text{C}(tʻtʻ\text{Indiv})\). Also \[ 1(tʻ\text{Indiv}) \in \text{N}_{0}\text{C}(tʻtʻ\text{Indiv}). \] And in the case supposed, \(1(tʻ\text{Indiv})\) is the maximum of \(\text{N}^{1}\text{C}(tʻtʻ\text{Indiv})\), but \(2(tʻ\text{Indiv}) \in\text{N}_{0}\text{C}(tʻtʻ\text{Indiv})\). Hence \[ \text{N}_{0}\text{C}(tʻtʻ\text{Indiv}) - \text{N}^{1}\text{C}(tʻtʻ\text{Indiv}) = \iota ʻ2. \] Generalizing, we see that \(\text{N}^{1}\text{C}(tʻ\alpha)\) consists of the same numbers as \(\text{N}_{0}\text{C}(\alpha)\) each raised one degree in type. Similar propositions hold of \(\text{N}^{2}\text{C}(t^{2}ʻ\alpha)\), [Pg 10]\(\text{N}^{3}\text{C}(t^{3}ʻ\alpha)\) etc.

It is often useful to have a notation for what we may call "the same cardinal in another type." Suppose \(\mu\) is a typically definite cardinal; then we will denote by \(\mu ^{(1)}\) the same cardinal in the next type, i.e. \[ \text{ sm }ʻʻ\mu \cap tʻ\mu . \] Note that, if \(\mu\) is a cardinal, \(\text{ sm }ʻʻ\mu \cap \mu = \mu\); and whether \(\mu\) is a typically definite cardinal or not, \[ \text{ sm }ʻʻ\mu \cap tʻ\alpha \] is a cardinal in a definite type. If \(\mu\) is typically definite, then \(\text{ sm }ʻʻ\mu \cap tʻ\alpha\) is wholly definite; if \(\mu\) is typically ambiguous, \(\text{ sm }ʻʻ\mu \cap tʻ\alpha\) has the same kind of indefiniteness as belongs to \(\text{NC}(\alpha)\). The most important case is when \(\mu\) is typically definite and \(\alpha\) has an assigned relation of type to \(\mu\). We then put, as observed above, \[ \begin{aligned} \mu ^{(1)} &= \text{ sm }ʻʻ\mu \cap tʻ\mu &&\quad\text{Df}\\ \mu ^{(2)} &= \text{ sm }ʻʻ\mu \cap t^{2}ʻ\mu &&\quad\text{Df etc}. \end{aligned} \] If \(\mu\) is an \(\text{N}_{0}\text{C}\), \(\mu ^{(1)}\) is an \(\text{N}^{1}\text{C}\) and \(\mu ^{(2)}\) is an \(\text{N}^{2}\text{C}\) and so on. \(\text{N}^{1}\text{C}(tʻ\alpha)\) will consist of all numbers which are of the form \(\mu^{(1)}\) for some \(\mu\) which is a member of \(\text{N}_{0}\text{C}(\alpha)\); i.e. \[ \text{N}^{1}\text{C}(tʻ\alpha) = \hat{\nu}\{(\exists \mu). \mu \in \text{N}_{0}\text{C}(\alpha) . \nu = \mu^{(1)}\}. \]

The second kind of non-homogeneous cardinals to be considered is called the class of "descending cardinals." These are such as go into a lower type; i.e. \(\text{Nc}(\alpha)ʻ\beta\) is a descending cardinal if \(\alpha\) is of a lower type than \(\beta\). We put \[ \begin{aligned} \text{N}_{1}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{1}ʻ\alpha &&\quad\text{Df}\\ \text{N}_{2}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{2}ʻ\alpha &&\quad\text{Df etc}.\\ \text{N}_{1}\text{C} &= \text{D}ʻ\text{N}_{1}\text{C} &&\quad\text{Df}\\ \text{N}_{2}\text{C} &= \text{D}ʻ\text{N}_{2}\text{C} &&\quad\text{Df etc}.\\ \mu_{(1)} &= \text{ sm }ʻʻ\mu \cap t_{1}ʻ\mu &&\quad\text{Df}\\ \mu_{(2)} &= \text{ sm }ʻʻ\mu \cap t_{2}ʻ\mu &&\quad\text{Df etc}. \end{aligned} \]

We have obviously \(\text{N}_{0}\text{c}ʻ\alpha = \text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha\).

Hence \(\text{N}_{0}\text{C}(\alpha) \subset\text{N}_{1}\text{C}(\alpha)\).

Also \(\gamma \in \text{N}_{1}\text{c}ʻ\delta . \supset .\text{N}_{1}\text{c}ʻ\delta = \text{N}_{0}\text{c}ʻ\gamma\), whence \(\exists !\text{N}_{1}\text{c}ʻ\delta . \supset .\text{N}_{1}\text{c}ʻ\delta \in \text{N}_{0}\text{C}\), whence \(\text{N}_{1}\text{C} - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}\).

Since also \(\Lambda {\sim} \in\text{N}_{0}\text{C}\text{N}_{0}\text{C}(\alpha)\), we find \[ \text{N}_{0}\text{C} = \text{N}_{1}\text{C} - \iota ʻ\Lambda, \] this proposition not requiring any further typical definiteness, since it holds however such definiteness may be introduced, remembering that such definiteness is necessarily so introduced as to secure significance. Further, in virtue of the fact that no class contained in \(t_{0}ʻ\alpha\) is similar to \(tʻ\alpha\), we have \[ \Lambda \in \text{N}_{1}\text{C}(\alpha). \][Pg 11] Consequently \(\text{N}_{1}\text{C} = \text{N}_{0}\text{C} \cup\iotaʻ\Lambda\).

We can prove in just the same way \[ \begin{aligned} \text{N}_{2}\text{C} &= \text{N}_{0}\text{C} \cup \iotaʻ\Lambda.\\ \text{Hence}\qquad \text{N}_{1}\text{C} &= \text{N}_{2}\text{C}, \end{aligned} \] and this result can obviously be extended to all descending cardinals.

The third kind of non-homogeneous cardinals to be considered may be called "relational cardinals." They are those applicable to classes of relations having a given relation of type to a given class. Consider for example \(\text{Nc}ʻ\in_{\Delta}ʻ\kappa\). (We shall take this as the definition of the product of the numbers of the members of \(\kappa\).) Suppose now that \(\kappa\) consists of a single term: we want to be able to say \[ \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa = \text{Nc}ʻ\breve{\iota}ʻ\kappa . \] We have in this case, if \(\kappa = \iota ʻ\alpha\), \[ \in_{\Delta }ʻ\kappa = \downarrow \alphaʻʻ\alpha, \] and we know that \(\downarrow \alphaʻʻ\alpha\,\text{ sm }\,\alpha\). But if we put simply \[ \text{Nc}ʻ\downarrow \alphaʻʻ\alpha = \text{Nc}ʻ\alpha, \] our proposition, though not mistaken, requires care in interpretation. Just as we put \(\iota ʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha\), so we want a notation giving typical definiteness to the proposition \(\downarrow \alpha ʻʻ\alpha \in \text{Nc}ʻ\alpha\). This is provided as follows.

Using the notation of *64, put \[ \begin{aligned} \text{N}_{00}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{0}ʻ\alpha &&\quad\text{Df}\\ \text{N}_{0}^{1}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{0}^{1}ʻ\alpha &&\quad\text{Df etc}.\\ \text{N}_{00}\text{C} &= \text{D}ʻ\text{N}_{00}c &&\quad\text{Df}\\ \text{N}_{0}^{1}\text{C} &= \text{D}ʻ\text{N}_{0}^{1}c &&\quad\text{Df etc}.\\ \mu_{00} &= \text{ sm }ʻʻ\mu \cap tʻt_{00}ʻt_{1}ʻ\mu &&\quad\text{Df etc}. \end{aligned} \] Then we have, for example, \[ \downarrow \alphaʻʻ\alpha \subset t_{0}^{1}ʻ\alpha,\quad\textit{i.e.}\quad \downarrow \alphaʻʻ\alpha \in tʻt_{0}^{1}ʻ\alpha. \] Hence \(\downarrow \alphaʻʻ\alpha \in\text{N}_{0}^{1}\text{c}ʻ\alpha\), where \(\text{N}_{0}^{1}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{0}^{1}ʻ\alpha\).

Similarly \(x \in tʻ\alpha . \supset . \downarrow xʻʻ\alpha \in\text{N}_{00}\text{c}ʻ\alpha\).

Thus the above definitions give us what is required.

In order to complete our notation for types, we should need to be able to express the type of the domain or converse domain of \(R\), or of any relation whose domain and converse domain have respectively given relations of type to the domain and converse domain of \(R\). Thus we might put \[ \begin{aligned} d_{0}ʻR &= t_{0}ʻ\text{D}ʻR &&\quad\text{Df}\\ b_{0}ʻR &= t_{0}ʻ\text{ᗡ}ʻR &&\quad\text{Df} \end{aligned} \] ("b" appears here as "d" written backwards) \[ \begin{aligned} d_{00}ʻR &= tʻ(d_{0}ʻR \uparrow b_{0}ʻR) &&\quad\text{Df}\\ &= tʻR\\ d^{mn}ʻR &= tʻ(t^{m}ʻd_{0}ʻR \uparrow t^{n}ʻb_{0}ʻR) &&\quad\text{Df and so on}. \end{aligned} \]

[Pg 12]

This notation would enable us to deal with descending relational cardinals. But it is not required in the present work, and is therefore not introduced among the numbered propositions.

When a typically ambiguous symbol, such as "\(\text{sm}\)" or "\(\text{Nc}\)," occurs more than once in a given context, it must not be assumed, unless required by the conditions of significance, that it is to receive the same typical determination in each case. Thus e.g. we shall write "\(\alpha\,\text{ sm }\, \beta . \supset .\beta\, \text{ sm }\, \alpha\)," although, if \(\alpha\) and \(\beta\) are of different types, the two symbols "\(\text{ sm }\)" must receive different typical determinations.

Formulae which are typically ambiguous, or only partially definite as to type, must not be admitted unless every significant interpretation is true. Thus for example we may admit \[ \unicode{x201c}\vdash . \alpha \in \text{Nc}ʻ\alpha\unicode{x201d} \] because here "\(\text{Nc}\)" must mean "\(\text{Nc}(\alpha_{\alpha})\)," so that the only ambiguity remaining is as to the type of \(\alpha\), and the formula holds whatever type \(\alpha\) may belong to, provided "\(\text{Nc}ʻ\alpha\)" is significant, i.e. provided \(\alpha\) is a class. But we must not, from "\(\alpha \in\text{Nc}ʻ\alpha\)," allow ourselves to infer \[ \unicode{x201c}\exists !\text{Nc}ʻ\alpha.\unicode{x201d} \] For here the conditions of significance no longer demand that "\(\text{Nc}\)" should mean "\(\text{Nc}(\alpha_{\alpha})\)": it might just as well mean "\(\text{Nc}(\beta_{\alpha })\)." And as we saw, if \(\beta\) is a lower type than \(\alpha\), and \(\alpha\) is sufficiently large of its type, we may have \[ \text{Nc}(\beta_{\alpha})ʻ\alpha = \Lambda, \] so that "\(\exists !\text{Nc}ʻ\alpha\)" is not admissible without qualification. Nevertheless, as we shall see in *100, there are a certain number of propositions to be made about a wholly ambiguous \(\text{Nc}\) or \(\text{NC}\).

Summary of *100.

In this number we shall be concerned only with such immediate consequences of the definition of cardinal numbers as do not require typical definiteness, beyond what the inherent conditions of significance may bestow. We introduce here the fundamental definitions:

*100·01. \(\text{Nc} = \overrightarrow{\text{ sm }} \quad\text{Df}\)

*100·02. \(\text{NC} = \text{D}ʻ\text{Nc} \quad\text{Df}\)

The definition "\(\text{Nc}\)" is required chiefly for the sake of the descriptive function \(\text{Nc}ʻ\alpha\). We have

*100·1. \(\vdash .\text{Nc}ʻ\alpha = \hat{\beta}(\beta\,\text{ sm }\,\alpha) = \hat{\beta}(\alpha\,\text{ sm }\,\beta)\)

This may be stated in various equivalent forms, which are given at the beginning of this number (*100·1—·16). After a few propositions on \(\text{Nc}\) as a relation, we proceed to the elementary properties of \(\text{Nc}ʻ\alpha\). We have

*100·3. \(\vdash .\alpha \in \text{Nc}ʻ\alpha\)

*100·31. \(\vdash :\alpha \in \text{Nc}ʻ\beta . \equiv . \beta \in \text{Nc}ʻ\alpha . \equiv . \alpha \text{ sm } \beta\)

*100·321. \(\vdash : \alpha\,\text{ sm }\,\beta . \supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta\)

*100·33. \(\vdash : \exists !\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta . \supset . \alpha\,\text{ sm }\,\beta\)

We proceed next to the elementary properties of \(\text{NC}\). We have

*100·4. \(\vdash : \mu \in \text{NC} . \equiv . (\exists \alpha ) . \mu = \text{Nc}ʻ\alpha\)

*100·42. \(\vdash : \mu ,\nu \in \text{NC} . \exists !\mu \cap \nu . \supset . \mu =\nu\)

*100·45. \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{Nc}ʻ\alpha = \mu \)

*100·51. \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{ sm }ʻʻ\mu = \text{Nc}ʻ\alpha\)

Observe that when we have such a hypothesis as "\(\mu \in \text{NC}\)," the \(\mu\), though it may be of any type, must be of some type; hence the \(\mu\) cannot have the typical ambiguity which belongs to \(\text{Nc}ʻ\alpha\). If we put \(\mu = \text{Nc}ʻ\alpha\), this will hold only in the type of \(\mu\); but "\(\text{ sm }ʻʻ\mu\)" is a typically ambiguous symbol, which[Pg 14] will represent in any type the "same" number as \(\mu\). Thus "\(\text{ sm }ʻʻ\mu = \text{Nc}ʻ\alpha\)" is an equation which is applicable to all possible typical determinations of "\(\text{ sm }\)" and "\(\text{Nc}\)."

*100·52. \(\vdash :\mu \in \text{NC}.\exists !\mu .\supset .\text{ sm }ʻʻ\mu \in \text{NC}\)

The hypothesis \(\exists !\mu\) is unnecessary, but we cannot prove this till later (*102).

We end the number with some propositions (*100·6—·64) stating that various classes (such as \(\iota ʻʻ\alpha\)), which have already been proved to be similar to \(\alpha\), have \(\text{Nc}ʻ\alpha\) members.

*100·01. \(\text{Nc}=\overrightarrow{\text{ sm }} \quad\text{Df}\)

*100·02. \(\text{NC}=\text{D}ʻ\text{Nc} \quad \text{Df}\)

*100·1. \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta}(\beta\, \text{ sm }\,\alpha)=\hat{\beta}(\alpha\, \text{ sm }\,\beta) \quad[*32·13.*73·31.(*100·01)]\)

*100·11. \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta}\{(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=\beta\} \quad[*100·1.*73·1]\)

*100·12. \(\vdash . \text{Nc}ʻ\alpha =\hat{\beta}\{(\exists R).R\in 1\rightarrow 1.\alpha \subset \text{D}ʻR.\beta =\breve{R}ʻʻ\alpha\} \quad[*100·1.*73·11]\)

*100·13. \(\vdash .\text{Nc}ʻ\alpha =\text{ᗡ}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{D}}ʻ\alpha )=\text{D}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{ᗡ}}ʻ\alpha)\)

Dem. \[ \begin{array}{l} \vdash .*100·11.*33·6. \supset \vdash .\text{Nc}ʻ\alpha &=\hat{\beta} \{(\exists R).R\in 1\rightarrow 1.R\in \overleftarrow{\text{D}}ʻ\alpha .\text{ᗡ}ʻR=\beta\}\\ [*22·33.*37·6] &=\text{ᗡ}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{D}}ʻ\alpha) &\qquad \text{(1)}\\ \vdash .*100·1.*73·1.*33·61. \supset \vdash .\text{Nc}ʻ\alpha &=\hat{\beta} \{(\exists R).R\in 1\rightarrow 1.R\in \overleftarrow{\text{ᗡ}}ʻ\alpha .\text{D}ʻR=\beta\}\\ [*22·33.*37·6] &=\text{D}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{ᗡ}}ʻ\alpha) &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*100·14. \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta} \{(\exists R).\alpha \subset \text{ᗡ}ʻR.R\upharpoonright \alpha \in 1\rightarrow 1.\beta = Rʻʻ\alpha\} \quad [*73·15.*100·1]\)

*100·15. \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta} \{(\exists R):\text{E}!!Rʻʻ\alpha : x,y\in \alpha .Rʻx=Rʻy.\supset _{x,y}.x=y:\beta =Rʻʻ\alpha\}\)

Dem. \[ \begin{array}{l} \vdash .*74·1·11.\supset \\ \vdash \colon\ldotp \text{E}!!Rʻʻ\alpha :x,y\in \alpha .&Rʻx=Rʻy.\supset _{x,y}.x=y:\beta =Rʻʻ\alpha :\equiv :\\ &R\upharpoonright \alpha \in 1\rightarrow \text{Cls}.\alpha \subset \text{ᗡ}ʻR.R\upharpoonright \alpha \in 1\rightarrow 1.\beta = Rʻʻ\alpha \qquad \text{(1)}\\ \vdash .(1).*4·71.*100·14.\supset \vdash .\text{Prop} \end{array} \]

*100·16. \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta} \{(\exists R)\colon\ldotp x,y\in \alpha .\supset _{x,y}:Rʻx=Rʻy.\equiv .x=y\colon\ldotp \beta =Rʻʻ\alpha\}\)

Dem. \[ \begin{array}{l} \vdash .*71·59.\supset \\ \vdash \colon\colon x,y\in \alpha .\supset_{x,y}:Rʻx=Rʻy.\equiv .x=y\colon\ldotp \equiv .R\upharpoonright \alpha \in 1\rightarrow 1.\alpha \subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ \vdash .(1).*100·14.\supset \vdash .\text{Prop} \end{array} \]

[Pg 15]

*100·2. \(\vdash .\text{E}!\text{Nc}ʻ\alpha \quad[*32·12.(*100·01)]\)

*100·21. \(\vdash .\text{ᗡ}ʻ\text{Nc} = \text{Cls}\)

Dem. \[ \begin{array}{l} &\vdash .*37·76.(*100·01).\supset \vdash .\text{ᗡ}ʻ\text{Nc}\subset \text{Cls} &\qquad \text{(1)}\\ &\vdash .*33·431.*100·2. \supset \vdash .\text{Cls}\subset \text{ᗡ}ʻ\text{Nc} &\qquad \text{(2)}\\ &\vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*100·22. \(\vdash .\text{Nc}\in 1\rightarrow \text{Cls} \quad[*72·12.(*100·01)]\)

*100·3. \(\vdash .\alpha \in \text{Nc}ʻ\alpha \quad[*73·3.*100·1]\)

Note that it is fallacious to infer \(\exists !\text{Nc}ʻ\alpha\), for reasons explained in the introduction to the present section.

*100·31. \(\vdash :\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta \quad[*32·18.*73·31.(*100·01)]\)

*100·32. \(\vdash :\alpha \in \text{Nc}ʻ\beta .\beta \in \text{Nc}ʻ\gamma .\supset .\alpha \in \text{Nc}ʻ\gamma \quad[*100·31.*73·32]\)

*100·321. \(\vdash :\alpha\,\text{ sm }\,\beta .\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*73·37.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\gamma\,\text{ sm }\,\alpha .\equiv _{\gamma }.\gamma \text{ sm }\beta :\\ [*100·1] &\supset :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

Note that \(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\supset .\alpha\,\text{ sm }\,\beta\) is not always true. We might be tempted to prove it as follows: \[ \begin{array}{l} \vdash .*100·1.\supset \vdash \colon\ldotp \text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\equiv :\gamma \,\text{ sm }\,\alpha .&\equiv _{\gamma }.\gamma \,\text{ sm }\,\beta :\\ [*10·1] &\supset :\alpha \,\text{ sm }\,\alpha .\equiv .\alpha \,\text{ sm }\,\beta :\\ [*73·3] &\supset :\alpha \,\text{ sm }\,\beta \end{array} \]

But the use of *10·1 here is only legitimate when the "\(\text{sm}\)" concerned is a homogeneous relation. If \(\text{Nc}ʻ\alpha\), \(\text{Nc}ʻ\beta\) are descending cardinals, we may have \(\text{Nc}ʻ\alpha =\Lambda = \text{Nc}ʻ\beta\) without having \(\alpha\text{ sm }\beta\).

*100·33. \(\vdash :\exists !\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta .\supset .\alpha \,\text{ sm }\,\beta\)

Dem. \[ \begin{array}{l} \vdash .*100·1.\supset \vdash :\text{Hp}.&\supset .(\exists \gamma ).\gamma\,\text{ sm }\, \alpha .\gamma \,\text{ sm }\, \beta.\\ [*73·31] &\supset .(\exists \gamma ).\alpha\, \text{ sm }\, \gamma .\gamma \text{ sm } \beta.\\ [*73·32] &\supset .\alpha\,\text{ sm }\, \beta :\supset \vdash .\text{Prop} \end{array} \]

Note that we do not always have \[ \alpha \,\text{ sm }\, \beta .\supset .\exists !\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta. \]

For if the \(\text{Nc}\) concerned is a descending \(\text{Nc}\), and \(\alpha\) and \(\beta\) are sufficiently great, \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) may both be \(\Lambda\). For example, we have \[ \text{Cl}ʻ(\alpha \cup - \alpha)\,\text{ sm }\, \text{Cl}ʻ(\alpha \cup - \alpha). \]

But \(\text{Nc}(\alpha)ʻ\text{Cl}ʻ(\alpha \cup - \alpha) = \Lambda\), so that \[ {\sim}\exists !\text{Nc}(\alpha )ʻ\text{Cl}ʻ(\alpha \cup - \alpha ) \cap \text{Nc}(\alpha )ʻ\text{Cl}ʻ(\alpha \cup - \alpha). \]

[Pg 16]

Thus "\(\alpha\, \text{ sm }\,\beta . \supset . \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta\)" is not always true when it is significant.

*100·34. \(\vdash : \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta . \supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta \quad[*100·33·321]\)

*100·35. \[\begin{aligned}\vdash \colon\ldotp \exists ! \text{Nc}ʻ\alpha . \lor . &\exists ! \text{Nc}ʻ\beta : \supset:\\ &\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . \equiv . \alpha \in \text{Nc}ʻ\beta . \equiv . \beta \in \text{Nc}ʻ\alpha . \equiv . \alpha \,\text{ sm }\, \beta \end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*22·5. \supset \vdash \colon\ldotp \text{Hp} . \supset : \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . &\supset . \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta.\\ [*100·33] &\supset . \alpha \, \text{ sm }\, \beta &\qquad \text{(1)}\\ \vdash .(1).*100·321. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . \equiv . \alpha \,\text{ sm }\, \beta &\qquad\text{(2)}\\ \vdash .(2).*100·31. &\supset \vdash . \text{Prop} \end{array} \]

Thus the only case in which the implications in *100·321·33·34 cannot be turned into equivalences is the case in which \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are both \(\Lambda\).

*100·36. \(\vdash \colon\ldotp \beta \in \text{Nc}ʻ\alpha . \supset : \exists ! \alpha . \equiv . \exists !\beta \quad[*100·31.*73·36]\)

*100·4. \(\vdash : \mu \in \text{NC} . \equiv . (\exists \alpha) . \mu = \text{Nc}ʻ\alpha \quad[*37·78·79.(*100·02·01)]\)

*100·41. \(\vdash . \text{Nc}ʻ\alpha \in \text{NC} \quad[*100·4·2.*14·204]\)

*100·42. \(\vdash : \mu,\nu \in \text{NC} . \exists !\mu \cap \nu . \supset . \mu = \nu\)

Dem. \[ \begin{array}{l} \vdash . *100·4 . \supset \vdash : \text{Hp} . &\supset . (\exists \alpha ,\beta ) . \mu = \text{Nc}ʻ\alpha . \nu = \text{Nc}ʻ\beta . \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ [*100·34] &\supset . (\exists \alpha ,\beta) . \mu = \text{Nc}ʻ\alpha . \nu = \text{Nc}ʻ\beta . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta.\\ [*14·15] &\supset . \mu = \nu : \supset \vdash . \text{Prop} \end{array} \]

*100·43. \(\vdash . \text{NC} \in \text{Cls}^{2} \text{excl} \quad[*100·42.*84·11]\)

*100·44. \(\vdash \colon\ldotp \mu \in \text{NC}. \exists ! \text{Nc}ʻ\alpha . \supset : \alpha \in \mu . \equiv . \text{Nc}ʻ\alpha = \mu\)

Dem. \[ \begin{array}{l} \vdash . *100·3. &\supset \vdash : \text{Nc}ʻ\alpha = \mu . \supset . \alpha \in \mu &\qquad\text{(1)}\\ \vdash . *10·24. \supset \vdash : &\mu \in \text{NC} . \exists ! \text{Nc}ʻ\alpha . \alpha \in \mu . \supset.\\ &\mu \in \text{NC} . \exists !\mu . \exists ! \text{Nc}ʻ\alpha . \alpha \in \mu.\\ [*100·4] &\supset . (\exists \beta ) . \mu = \text{Nc}ʻ\beta . \exists ! \text{Nc}ʻ\beta. \exists ! \text{Nc}ʻ\alpha . \alpha \in \text{Nc}ʻ\beta.\\ [*100·35] &\supset . (\exists \beta). \mu = \text{Nc}ʻ\beta . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta.\\ [*14·15] &\supset . \text{Nc}ʻ\alpha = \mu &\qquad \text{(2)}\\ \vdash . (1).(2). &\supset \vdash . \text{Prop} \end{array} \]

*100·45. \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{Nc}ʻ\alpha = \mu \quad[*100·4·31·321]\)

*100·5. \(\vdash : \mu \in \text{NC} . \alpha ,\beta \in \mu . \supset . \alpha \,\text{ sm }\, \beta\)

Dem. \[ \begin{array}{l} \vdash . *100·4. \supset \vdash : \text{Hp} . &\supset . (\exists \gamma ) . \mu = \text{Nc}ʻ\gamma . \alpha ,\beta \in \text{Nc}ʻ\gamma.\\ [*100·31] &\supset .(\exists \gamma). \alpha\,\text{sm}\,\gamma . \beta\,\text{ sm }\,\gamma.\\ [*73·31·32] &\supset . \alpha\,\text{ sm }\,\beta : \supset \vdash . \text{Prop} \end{array} \]

[Pg 17]

*100·51. \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{ sm }ʻʻ\mu = \text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash. *100·5 . \text{Fact}. \supset \vdash \colon\ldotp \text{Hp} . \supset : \beta \in \mu . \gamma \,\text{ sm }\, \beta . &\supset . \alpha \text{ sm } \beta . \gamma \text{ sm } \beta.\\ [*73·31·32] &\supset . \alpha \,\text{ sm }\, \gamma.\\ [*100·31] &\supset . \gamma \in \text{Nc}ʻ\alpha &\qquad\text{(1)}\\ \vdash .(1).*10·11·21·23.*37·1. \supset \vdash : \text{Hp} . &\supset . \text{ sm }ʻʻ\mu \subset \text{Nc}ʻ\alpha &\qquad \text{(2)}\\ \vdash .*100·31. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \gamma \in \text{Nc}ʻ\alpha . \supset . \gamma \,\text{ sm }\, \alpha . \alpha \in \mu.\\ [*37·1] &\supset . \gamma \in \text{ sm }ʻʻ\mu &\qquad \text{(3)}\\ \vdash .(2).(3). \supset \vdash . \text{Prop} \end{array} \]

*100·511. \(\vdash : \exists ! \text{Nc}ʻ\beta . \supset . \text{ sm }ʻʻ\text{Nc}ʻ\beta = \text{Nc}ʻ\beta\)

Here the last "\(\text{Nc}ʻ\beta\)" may be of a different type from the others: the proposition holds however its type is determined.

Dem. \[ \begin{array}{l} \vdash .*100·51·41. \supset \vdash : \alpha \in \text{Nc}ʻ\beta . \supset . \text{ sm }ʻʻ\text{Nc}ʻ\beta &= \text{Nc}ʻ\alpha \\ [*100·31·321] &= \text{Nc}ʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*10·11·23. \supset \vdash . \text{Prop} \end{array} \]

*100·52. \(\vdash : \mu \in \text{NC} . \exists ! \mu . \supset . \text{ sm }ʻʻ\mu \in \text{NC} \quad[*100·51·4]\)

This proposition still holds when \(\mu = \Lambda\), but the proof is more difficult, since it depends upon the proof that every null-class of classes is an \(\text{NC}\), which in turn depends upon the proof that \(\text{Cl}ʻ\alpha\) is not similar to \(\alpha\) or to any class contained in \(\alpha\).

*100·521. \(\vdash : \mu \in \text{NC} . \exists ! \text{ sm }ʻʻ\mu . \supset . \text{ sm }ʻʻ\text{ sm }ʻʻ\mu = \mu\)

Dem. \[ \begin{array}{l} \vdash .*37·29.\text{Transp}. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \exists !\mu:\\ [*100·52] &\supset : \text{ sm }ʻʻ\mu \in \text{NC}:\\ [*100·51.\text{Hp}] &\supset : \gamma \in \text{ sm }ʻʻ\mu . \supset . \text{ sm }ʻʻ\text{ sm }ʻʻ\mu = \text{Nc}ʻ\gamma &\qquad \text{(1)}\\ \vdash .*37·1.\text{Fact}. \supset \vdash : \text{Hp} . \gamma \in \text{ sm }ʻʻ\mu . &\supset . (\exists \alpha) . \alpha \in \mu . \mu \in \text{NC} . \gamma \text{ sm } \alpha.\\ [*100·45·321] &\supset . (\exists \alpha) . \text{Nc}ʻ\alpha = \mu . \text{Nc}ʻ\gamma = \text{Nc}ʻ\alpha.\\ [*13·17] &\supset . \text{Nc}ʻ\gamma = \mu &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp} . \gamma \in \text{ sm }ʻʻ\mu . \supset . \text{ sm }ʻʻ\text{ sm }ʻʻ\mu = \mu &\qquad \text{(3)}\\ \vdash .(3).*10·11·23·35 . \supset \vdash . \text{Prop} \end{array} \]

*100·53. \(\vdash \colon\ldotp \exists !\mu . \exists !\nu . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \equiv . \nu \in \text{NC} . \mu = \text{ sm }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash .*100·52. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \supset . \nu \in \text{NC} &\qquad \text{(1)}\\ \vdash .*100·521. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \supset . \mu = \text{ sm }ʻʻ\nu &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp} . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \supset . \nu \in \text{NC} . \mu = \text{ sm }ʻʻ\nu &\qquad \text{(3)}\\ \vdash .(3).(3)\dfrac{\nu,\,\mu}{\mu,\,\nu}. \supset \vdash . \text{Prop} \end{array} \]

[Pg 18]

*100·6. \(\vdash .\iotaʻʻ\alpha \in \text{Nc}ʻ\alpha \quad[*73·41.*100·31]\)

*100·61. \(\vdash .\hat{\beta}\{(\exists y).y \in \alpha .\beta = \iota ʻx \cup \iota ʻy\} \in \text{Nc}ʻ\alpha \quad [*73·27.*54·21.*100·31]\)

*100·62. \(\vdash .x\downarrow ʻʻ\alpha \in \text{Nc}ʻ\alpha \quad[*73·61.*100·31]\)

*100·621. \(\vdash .\downarrow xʻʻ\alpha \in \text{Nc}ʻ\alpha \quad[*73·611.*100·31]\)

*100·63. \(\vdash .\in_{\Delta }ʻ\iota ʻ\alpha \in \text{Nc}ʻ\alpha \quad[*83·41.*100·31]\)

*100·631. \(\vdash .\text{D}ʻʻ{\in}_{\Delta}ʻ\iota ʻ\alpha \in \text{Nc}ʻ\alpha \quad[*83·7.*100·6]\)

*100·64. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl} .\supset . \text{D}ʻʻ\in_{\Delta }ʻ\kappa \subset \text{Nc}ʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*84·3.*80·14.\supset \vdash :\text{Hp}.R\in {\in}_{\Delta}ʻ\kappa .&\supset .R \in 1\rightarrow 1.\kappa = \text{ᗡ}ʻR.\\ [*73·2.*100·31] &\supset .\text{D}ʻR \in \text{Nc}ʻ\kappa :\supset \vdash . \text{Prop} \end{array} \]

Summary of *101.

In the present number, we have to show that 0 and 1 and 2 as previously defined are cardinal numbers in the sense defined in *100, and to add a few elementary propositions to those already given concerning them. We prove (*101·12·241) that 0 and 1 are not null, which cannot be proved, with our axioms, for any other cardinal, except (in the case of finite cardinals) when the type is specified as a sufficiently high one. Thus we prove (*101·42·43) that \(2_{\text{Cls}}\) and \(2_{\text{Rel}}\) exist; this follows from \(\Lambda \neq \text{V}\) and \(\dot{\Lambda}\neq \dot{\text{V}}\). We prove (*101·22·34) that 0 and 1 and 2 are all different from each other. We prove (*101·15·28) that \(\text{ sm }ʻʻ0 = 0\) and \(\text{ sm }ʻʻ1 = 1\), but we cannot prove \(\text{ sm }ʻʻ2 = 2\) unless we assume the existence of at least two individuals, or define the first 2 in "\(\text{ sm }ʻʻ2 = 2\)" as a 2 of some type other than \(2_{\text{Indiv}}\), where "\(\text{Indiv}\)" stands for the type of individuals.

It should be observed that, since 0 and 1 and 2 are typically ambiguous, their properties are analogous to those of "\(\text{Nc}ʻ\alpha\)" rather than to those of \(\mu\), where \(\mu \in\text{NC}\). For example, we have

*100·511. \(\vdash : \exists ! \text{Nc}ʻ\beta . \supset . \text{ sm }ʻʻ\text{Nc}ʻ\beta = \text{Nc}ʻ\beta\)

but we shall not have \(\mu \in \text{NC} . \exists ! \mu . \supset. \text{ sm }ʻʻ\mu =\mu\) unless the "\(\text{ sm }\)" concerned is homogeneous, since in other cases the symbols do not express a significant proposition. But in *100·511 we may substitute 0 or 1 or 2, and the proposition remains significant and true. In fact we have (*101·1·2·31) \[ \vdash . 0 = \text{Nc}ʻ\Lambda . 1 = \text{Nc}ʻ\iota ʻx . 2 = \text{Nc}ʻ(\iota ʻ\iota ʻx \cup \iota ʻ\Lambda ), \] where 0 and 1 and 2 have an ambiguity corresponding to that of "\(\text{Nc}\)."

*101·1. \(\vdash . 0 = \text{Nc}ʻ\Lambda \quad[*73·48 . *100·1]\)

*101·11. \(\vdash . 0 \in \text{NC} \quad[*101·1 .*100·4]\)

*101·12. \(\vdash . \exists ! 0 \quad[*51·161. (*54·01)]\)

*101·13. \(\vdash . \exists ! 0 \cap \text{Cl}ʻ\alpha . \Lambda \in 0 \cap \text{Cl}ʻ\alpha \quad[*51·16 . *60·3]\)

[Pg 20]

*101·14. \(\vdash : \text{Nc}ʻ\gamma = 0 . \equiv . \gamma = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*101·1·12 . \supset \vdash : \text{Nc}ʻ\gamma = 0 . &\equiv . \text{Nc}ʻ\gamma = \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\Lambda.\\ [*13·194] &\equiv . \text{Nc}ʻ\gamma = \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\gamma.\\ [*100·35] &\equiv . \gamma \in \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\gamma.\\ [*101·1.*54·102] &\equiv . \gamma = \Lambda . \exists ! \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\gamma .\\ [*101·1·12.*13·194] &\equiv . \gamma = \Lambda : \supset \vdash . \text{Prop} \end{array} \]

*101·15. \(\vdash . \text{ sm }ʻʻ0 = 0\)

Dem. \[ \begin{array}{l} \vdash .*37·1 . \supset \vdash : \gamma \in \text{ sm }ʻʻ0 . &\equiv . (\exists \alpha ) . \alpha \in 0 . \gamma \text{ sm } \alpha.\\ [*54·102] &\equiv . \gamma \text{ sm } \Lambda .\\ [*73·48] &\equiv . \gamma \in 0 : \supset \vdash . \text{Prop} \end{array} \]

*101·16. \(\vdash \colon\ldotp \mu \in \text{NC}-\iota ʻ0 . \supset : \alpha \in \mu . \supset_{\alpha} . \exists !\alpha\)

Dem. \[ \begin{array}{l} \vdash .*100·45 . \supset \vdash : \mu \in \text{NC} . \Lambda \in \mu . &\supset . \mu = \text{Nc}ʻ\Lambda\\ [*101·1] & =0 &\qquad \text{(1)}\\ \vdash .(1).\text{Transp}. &\supset \vdash \colon\ldotp \mu \in \text{NC}-\iota ʻ0 . \supset : \Lambda {\sim}\in \mu :\\ [*24·63] &\supset : \alpha \in \mu . \supset_{\alpha} . \exists !\alpha \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

*101·17. \(\vdash : \Lambda \in \text{Nc}ʻ\alpha . \equiv . \text{Nc}ʻ\alpha = 0 . \equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\Lambda . \equiv . \alpha = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*100·31·321. \supset \vdash : \Lambda \in \text{Nc}ʻ\alpha . &\supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\Lambda.\\ [*101·1] &\supset . \text{Nc}ʻ\alpha = 0 &\qquad \text{(1)}\\ \vdash .*101·13. &\supset \vdash : \text{Nc}ʻ\alpha = 0 . \supset . \Lambda \in \text{Nc}ʻ\alpha &\qquad\text{(2)}\\ \vdash .(1).(2). \supset \vdash : \Lambda \in \text{Nc}ʻ\alpha . &\equiv . \text{Nc}ʻ\alpha = 0. &\qquad\text{(3)}\\ [*101·1] &\equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\Lambda . &\qquad\text{(4)}\\ [*101·14] &\equiv . \alpha = \Lambda &\qquad \text{(5)}\\ \vdash .(3).(4).(5). \supset \vdash . \text{Prop} \end{array} \]

*101·2. \(\vdash . 1 = \text{Nc}ʻ\iota ʻx \quad[*73·45.*100·1]\)

*101·21. \(\vdash . 1 \in \text{NC} \quad[*101·2.*100·4]\)

*101·22. \(\vdash . 1 \neq 0\)

Dem. \[ \begin{array}{l} \vdash .*52·21.*101·13. &\supset \vdash . \Lambda {\sim}\in 1 . \Lambda \in 0.\\ [*13·14] &\supset \vdash . 1\neq 0 \end{array} \]

*101·23. \(\vdash . 1 \cap 0 = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*52·21. \supset \vdash : \alpha \in 1 . &\supset . \alpha \neq \Lambda.\\ [*54·102] &\supset . \alpha {\sim}\in 0 &\qquad\text{(1)}\\ \vdash .(1).*24·39. &\supset \vdash . \text{Prop} \end{array} \]

[Pg 21]

*101·24. \(\vdash : \exists !\alpha . \supset . \exists ! 1 \cap \text{Cl}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash . *52·22.*60·6 . \supset \vdash : x \in \alpha . \supset . \iota ʻx \in 1 \cap \text{Cl}ʻ\alpha \qquad \text{(1)}\\ \vdash .(1).*10·11·28. \supset \vdash . \text{Prop} \end{array} \]

*101·241. \(\vdash . \exists !1 \quad[*52·23]\)

*101·25. \(\vdash : \alpha \in 1 . \beta \subset \alpha . \beta \neq \alpha . \supset . \beta \in 0\)

Dem. \[ \begin{array}{l} \vdash .*52·64.*22·621. &\supset \vdash : \alpha \in 1 . \beta \subset \alpha . \supset . \beta \in 1 \cup 0 &\qquad\text{(1)}\\ \vdash .*52·46. &\supset \vdash : \alpha,\beta \in 1 . \beta \subset \alpha . \supset . \beta =\alpha:\\ [\text{Transp}] &\supset \vdash : \alpha \in 1 . \beta \subset \alpha . \beta \neq \alpha . \supset . \beta {\sim}\in 1 &\qquad\text{(2)}\\ \vdash .(1).(2). \supset \vdash . \text{Prop} \end{array} \]

*101·26. \(\vdash . sʻ\text{Cl}ʻʻ1 = 0 \cup 1\)

Dem. \[ \begin{array}{l} \vdash .*60·371.*40·43. &\supset \vdash . sʻ\text{Cl}ʻʻ1 \subset 0 \cup 1 &\qquad\text{(1)}\\ \vdash .*60·3·34. &\supset \vdash . \Lambda \in \text{Cl}ʻ\iota ʻx . \iota ʻx \in \text{Cl}ʻ\iota ʻx.\\ [*52·22.*40·4] &\supset \vdash . \Lambda \in sʻ\text{Cl}ʻʻ1 . \iota ʻx \in sʻ\text{Cl}ʻʻ1.\\ [*51·2.*52·1] &\supset \vdash . 0 \subset sʻ\text{Cl}ʻʻ1 . 1 \subset sʻ\text{Cl}ʻʻ1 &\qquad\text{(2)}\\ \vdash .(1).(2). \supset \vdash .\text{Prop} \end{array} \]

*101·27. \(\vdash . 1 = \hat{\alpha}\{(\exists x) . x \in \alpha . \alpha - \iota ʻx \in 0\}\)

Dem. \[ \begin{array}{l} \vdash .*54·102. \supset \vdash : (\exists x) . x \in \alpha . \alpha - \iota ʻx \in 0 . &\equiv . (\exists x) . x \in \alpha . \alpha - \iota ʻx = \Lambda .\\ [*24·3] &\equiv . (\exists x). x \in \alpha . \alpha \subset \iota ʻx.\\ [*51·2] &\equiv . (\exists x) . \alpha = \iota ʻx.\\ [*52·1] &\equiv . \alpha \in 1 : \supset \vdash .\text{Prop} \end{array} \]

*101·28. \(\vdash . \text{ sm }ʻʻ1 = 1\)

Dem. \[ \begin{array}{l} \vdash .*37·1. \supset \vdash : \gamma \in \text{ sm }ʻʻ1 . &\equiv . (\exists \alpha ) . \alpha \in 1 . \gamma\,\text{ sm }\,\alpha.\\ [*52·1] &\equiv . (\exists x) . \gamma\,\text{ sm }\,\iota ʻx.\\ [*73·45] &\equiv . \gamma \in 1 : \supset \vdash . \text{Prop} \end{array} \]

*101·29. \(\vdash : \iota ʻx\in \text{Nc}ʻ\alpha . \equiv . \text{Nc}ʻ\alpha = 1. \equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\iota ʻx. = . \alpha \in 1\)

Dem. \[ \begin{array}{l} \vdash .*100·31·321. \supset \vdash : \iota ʻx \in \text{Nc}ʻ\alpha . &\supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\iota ʻx.\\ [*101·2] &\supset . \text{Nc}ʻ\alpha = 1 &\qquad\text{(1)}\\ \vdash .*52·22. &\supset \vdash : \text{Nc}ʻ\alpha = 1 . \supset . \iota ʻx \in \text{Nc}ʻ\alpha &\qquad\text{(2)}\\ \vdash .(1).(2). \supset \vdash : \iota ʻx \in \text{Nc}ʻ\alpha . &\equiv . \text{Nc}ʻ\alpha = 1. &\qquad\text{(3)}\\ [*101·2] &\equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\iota ʻx &\qquad\text{(4)}\\ \vdash .*101·2.*52·1. &\supset \vdash : \alpha \in 1 . \supset . \text{Nc}ʻ\alpha = 1 &\qquad\text{(5)}\\ \vdash .*100·3 . &\supset \vdash : \text{Nc}ʻ\alpha = 1 . \supset . \alpha \in 1 &\qquad\text{(6)}\\ \vdash .(3).(4).(5).(6). \supset \vdash . \text{Prop} \end{array} \]

[Pg 22]

*101·3. \(\vdash :x\neq y.\supset .2=\text{Nc}ʻ(\iota ʻx\cup \iota ʻy)\)

Dem. \[ \begin{array}{l} \vdash .*73·71·43.*51·231.\supset \vdash \colon\ldotp \text{Hp}.&\supset :z\neq w.\supset .(\iota ʻz\cup \iota ʻw)\text{ sm }(\iota ʻx\cup \iota ʻy):\\ [*54·101]&\supset :\beta \in 2.\supset .\beta \text{ sm }(\iota ʻx\cup \iota ʻy):\\ [*100·1] &\supset :2\subset \text{Nc}ʻ(\iota ʻx\cup \iota ʻy) &\qquad \text{(1)}\\ \vdash .*53·32.*71·163.\supset \vdash :R\in 1\rightarrow 1.x,y&\in \text{ᗡ}ʻR.\supset .\\ &Rʻʻ(\iota ʻx\cup \iota ʻy)=\iota ʻRʻx\cup \iota ʻRʻy &\qquad \text{(2)}\\ \vdash .*71·56.\text{Transp}. &\supset \vdash :\text{Hp}.R\in 1\rightarrow 1.x,y\in \text{ᗡ}ʻR.\supset .Rʻx\neq Rʻy &\qquad \text{(3)}\\ \vdash .(2).(3).*54·26.&\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :R\in 1\rightarrow 1.x,y\in \text{ᗡ}ʻR.\beta =Rʻʻ(\iota ʻx\cup \iota ʻy).\supset .\beta \in 2:\\ [*10·11·21·23.*51·234]&\supset :(\exists R).R\in 1\rightarrow 1.\iota ʻx\cup \iota ʻy\subset \text{ᗡ}ʻR.\beta =Rʻʻ(\iota ʻx\cup \iota ʻy).\\ &\supset .\beta \in 2:\\ [*73·12.*100·1] &\supset :\text{Nc}ʻ(\iota ʻx\cup \iota ʻy)\subset 2 &\qquad \text{(4)}\\ \vdash .(1).(4).\supset \vdash .\text{Prop} \end{array} \]

*101·301. \(\vdash.2=\hat{\alpha}\{(\exists x).x\in \alpha .\alpha -\iota ʻx\in 1\} \quad[*54·3]\)

In comparing *101·31 with *101·1·2·3, it should be observed that \(\iota ʻx\) and \(\Lambda\) are both classes, whereas in *101·1·2·3 there was no typical limitation beyond what was imposed by the conditions of significance.

*101·31. \(\vdash .2=\text{Nc}ʻ(\iota ʻ\iota ʻx\cup \iota ʻ\Lambda)\)

Dem. \[ \begin{array}{l} \vdash .*51·161. &\supset \vdash .\iota ʻx\neq \Lambda &\qquad \text{(1)}\\ \vdash .(1).*101·3.&\supset \vdash .\text{Prop} \end{array} \]

*101·32. \(\vdash .2\in \text{NC} \quad[*101·31.*100·4]\)

*101·33. \(\vdash :\alpha,\beta \in 1.\alpha \cap \beta =\Lambda .\supset .\alpha \cup \beta \in 2 \quad[*54·43]\)

*101·34. \(\vdash .2\neq 0.2\neq 1\)

Dem. \[ \begin{array}{l} \vdash .*101·13.&\supset \vdash .\Lambda \in 0 &\qquad \text{(1)}\\ \vdash .*101·301. &\supset \vdash :\alpha \in 2.\supset .\exists !\alpha :\\ [*24·63] &\supset \vdash .\Lambda {\sim}\in 2 &\qquad \text{(2)}\\ \vdash .(1).(2).*13·14. &\supset \vdash .2\neq 0 &\qquad \text{(3)}\\ \vdash .*52·22.*54·26.*22·56. &\supset \vdash .\iota ʻy\in 1.\iota ʻy{\sim}\in 2.\\ [*13·14]&\supset \vdash .1\neq 2 &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*101·35. \(\vdash .2\cap 0=\Lambda .2\cap 1=\Lambda \quad[*100·42.\text{Transp}.*101·11·21·32·34]\)

*101·36. \(\vdash :\alpha \in 2.\beta \subset \alpha .\beta \neq \alpha .\supset .\beta \in 0\cup 1\)

Dem. \[ \begin{array}{l} \vdash .*54·42. &\supset \vdash :\alpha \in 2.\beta \subset \alpha .\exists !\beta .\beta \neq \alpha .\supset .\beta \in 1 &\qquad \text{(1)}\\ \vdash .*54·102. &\supset \vdash :{\sim}\exists !\beta .\supset .\beta \in 0 &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash .\text{Prop} \end{array} \]

[Pg 23]

*101·37. \(\vdash . sʻ\text{Cl}ʻʻ2 \subset 0\cup 1\cup 2 \quad[*54·411]\)

*101·38. \(\vdash : \exists ! 2 . \supset . sʻ\text{Cl}ʻʻ2 = 0\cup 1\cup 2\)

Dem. \[ \begin{array}{l} \vdash .*60·3. \supset \vdash : \text{Hp} . &\supset . (\exists \alpha) . \alpha \in 2 . \Lambda \in \text{Cl}ʻ\alpha.\\ [*40·4] &\supset . \Lambda \in sʻ\text{Cl}ʻʻ2.\\ [*51·2] &\supset . 0 \subset sʻ\text{Cl}ʻʻ2 &\qquad\text{(1)}\\ \vdash .*60·34.&\supset \vdash . 2 \subset sʻ\text{Cl}ʻʻ2 &\qquad\text{(2)}\\ \vdash .*54·101. \supset \vdash \colon\colon \text{Hp} . &\supset \colon\ldotp (\exists x,y) . x \neq y \colon\ldotp \\ [*13·171.\text{Transp}] &\supset \colon\ldotp (\exists x,y) \colon\ldotp (z) : z\neq x . \lor . z\neq y\colon\ldotp\\ [*54·26] &\supset \colon\ldotp (\exists x,y) \colon\ldotp (z) : \iota ʻz \cup \iota ʻx \in 2 . \lor . \iota ʻz \cup \iota ʻy \in 2\colon\ldotp\\ [*11·26.*22·58] &\supset \colon\ldotp (z) \colon\ldotp (\exists \alpha ,\beta ) : \alpha \in 2 . \iota ʻz \in \text{Cl}ʻ\alpha . \lor . \beta \in 2 . \iota ʻz \in \text{Cl}ʻ\beta \colon\ldotp \\ [*40·4] &\supset \colon\ldotp (z) . \iota ʻz \in sʻ\text{Cl}ʻʻ2\colon\ldotp \\ [*52·1] &\supset \colon\ldotp 1 \subset sʻ\text{Cl}ʻʻ2 &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*101·37 . \supset \vdash . \text{Prop} \end{array} \]

*101·4. \(\vdash : (\exists x,y) . x\neq y . \equiv . \exists !2\)

Dem. \[ \begin{array}{l} \vdash . *54·26 . &\supset \vdash : x\neq y . \supset . \exists !2:\\ [*11·11·35] &\supset \vdash : (\exists x,y) . x\neq y . \supset . \exists !2 &\qquad \text{(1)}\\ \vdash .*54·101. &\supset \vdash : \alpha \in 2 . \supset . (\exists x,y) . x\neq y:\\ [*10·11·23] &\supset \vdash : \exists !2 . \supset . (\exists x,y) . x\neq y &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash . \text{Prop} \end{array} \]

When we are considering the lowest type occurring in a context, our premisses do not suffice to prove \((\exists x,y).x\neq y\). For every other type, this can be proved. Thus \(\Lambda \neq \text{V}\) and \(\dot{\Lambda} \neq \dot{\text{V}}\) give the required result for classes and relations respectively.

*101·41. \(\vdash : (\exists x) . \iota ʻx\neq \text{V}. \equiv . \exists !2\)

Dem. \[ \begin{array}{l} \vdash .*24·14.\text{Transp}. \supset \\ \vdash \colon\ldotp (\exists x) . \iota ʻx\neq \text{V} . &\equiv : (\exists x) : (\exists y) . y {\sim}\in \iota ʻx :\\ [*51·15] &\equiv : (\exists x,y) . x\neq y:\\ [*101·4] &\equiv : \exists !2 \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

*101·42. \(\vdash . \exists ! 2_{\text{Cls}} . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2_{\text{Cls}}\)

Dem. \[ \begin{array}{l} \vdash .*20·41.*24·1. &\supset \vdash . \Lambda , \text{V} \in \text{Cls} . \Lambda \neq \text{V} &\qquad \text{(1)}\\ \vdash .(1).*54·26. &\supset \vdash . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2 . \iota ʻ\Lambda \cup \iota ʻ\text{V} \subset \text{Cls}.\\ [*63·371·105] &\supset \vdash . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2 \cap tʻ\text{Cls}.\\ [(*65·01)] &\supset \vdash . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2_{\text{Cls}} . \supset \vdash . \text{Prop} \end{array} \]

*101·43. \(\vdash . \exists ! 2_{\text{Rel}} \quad[\text{Proof as in *101·42}]\)

Summary of *102.

In this number, we shall consider a typically definite relation "\(\text{Nc}\)," i.e. we shall consider the relation, to a class \(\delta\) which is given as of the same type as \(\beta\), of the class \(\mu\) of those classes \(\gamma\) which are similar to \(\delta\) and of the same type as \(\alpha\). We shall then put \[ \begin{aligned} \mu &= \text{Nc}(\alpha _{\beta })ʻ\delta ,\\ \gamma &\in \text{Nc}(\alpha _{\beta })ʻ\delta ,\\ \gamma &\,\text{ sm }_{(\alpha ,\beta )}\delta , \end{aligned} \] and the class of all such numbers as \(\mu\) for a given \(\alpha\) and \(\beta\) we shall call \(\text{NC}^{\beta }(\alpha)\), so that \[ \text{NC}^{\beta }(\alpha ) = \text{D}ʻ\text{Nc}(\alpha _{\beta }). \]

The notations here introduced for giving typical definiteness to "\(\text{sm}\)" and "\(\text{Nc}\)" are those defined in *65 for any typically ambiguous relation.

By *63·01·02 we have, if \(\alpha\) is a typically ambiguous symbol, \[ \begin{aligned} &\vdash .\alpha _{x} = \alpha \cap tʻx,\\ &\vdash .\alpha (x) = \alpha \cap tʻ\iota ʻx. \end{aligned} \]

Thus \(\vdash .\alpha (x) = \alpha _{\iota ʻx}\). If we apply the definitions to 1, "\(1_{x}\)" is meaningless unless \(x\) is a class; we therefore write a Greek letter in place of \(x\), and we have \[ \vdash .1_{\beta } = 1\cap tʻ\beta = 1\cap (\iota ʻ\beta \cup -\iota ʻ\beta ). \]

If \(x\in \beta\), we shall have \(\iota ʻx = \beta .\lor.\iota ʻx\neq\beta\). Hence \[ \begin{aligned} &\vdash :x\in \beta .\supset .\iota ʻx\in 1_{\beta }.\\ \text{Similarly}\qquad &\vdash :x{\sim}\in \beta .\supset .\iota ʻx\in 1_{\beta }.\\ \text{Thus}\qquad &\vdash :x\in t_{0}ʻ\beta .\supset .\iota ʻx\in 1_{\beta }. \end{aligned} \]

The converse implication also holds, so that \[ \vdash :x\in t_{0}ʻ\beta .\equiv .\iota ʻx\in 1_{\beta }. \]

Thus \(1_{\beta}\) consists of all unit classes whose sole members \(x\) either are or are not members of \(\beta\), i.e. for which "\(x\in \beta\)" is significant.

In "\(x\in t_{0}ʻ\beta .\supset .\iota ʻx\in 1_{\beta }\)," the hypothesis renders explicit the condition of significance; thus "\(\iota ʻx\in 1_{\beta }\)" is always true when significant, and always significant when \(x\in t_{0}ʻ\beta\). On the interpretation of negative statements concerning types, see the note at the end of this number.

It should be noted that all the constant relations introduced in this work are typically ambiguous. Consider e.g. \(\dot{\Lambda}\), \(\text{sg}\), \(\text{D}\), \(s\), \(\dot{s}\), \(I\), \(\iota\), \(\in\), \(\text{Cl}\), \(\text{Rl}\). These[Pg 25] all have more or less typical ambiguity, though all of them have what we will call relative typical definiteness, i.e. when the type of the relatum is given, that of the referent is given also. (In regard to \(\text{D}\), it is not true that, conversely, when the type of the referent is given, that of the relatum is also given.) But "\(\text{sm}\)" and "\(\text{Nc}\)" have not even relative definiteness. When the type of the relatum is given, that of the referent becomes no more definite than before; the only restrictions are that the relatum for "\(\text{sm}\)" or "\(\text{Nc}\)" must be a class, that the referent for "\(\text{sm}\)" must be a class, and that the referent for "\(\text{Nc}\)" must be a class of classes. When a relation \(R\) has relative definiteness, it is enough to fix the type of the relatum; and if further \(R\in 1\rightarrow \text{Cls}\), so that \(R\) leads to a descriptive function, "\(Rʻy\)" has complete typical definiteness as soon as the type of \(y\) is given. Now the constant relations hitherto introduced, with the exception of "\(\text{sm}\)" and "\(\dot{\text{V}}\)," have all been one-many relations, and have been used almost exclusively in the form of descriptive functions. Hence no special notation has been required to give typical definiteness, since "\(Rʻy\)" in these circumstances, has typical definiteness as soon as \(y\) is assigned. But with the consideration of "\(\text{sm}\)" and "\(\text{Nc}\)," which do not have even relative definiteness, an explicit means of giving typical definiteness becomes necessary. It should be observed, however, that "\(\text{Nc}ʻ\delta\)" has typical definiteness, when \(\delta\) is known, as soon as the domain of "\(\text{Nc}\)" has typical definiteness, since \(\delta\) must belong to the converse domain. It is for the sake of this and similar cases that we introduced the two definitions in *65, which only give typical definiteness to the domain.

In virtue of the definitions in *65, if \(R\) is a typically ambiguous relation, and \(x\) is a referent, \(R\) becomes \(R_{x}\); if, further, \(y\) is a relatum, \(R\) becomes \(R_{(x,y)}\). If \(x\) is a referent for \(R\), we have \((\exists y).x\in \overrightarrow{R}ʻy\), and \(\overrightarrow{R}ʻy\in \text{D}ʻ\overrightarrow{R}\). Thus \(\text{D}ʻ\overrightarrow{R}\) has a member of the type next above that of \(x\), i.e. of the type of \(\iota ʻx\). Thus \[ \begin{aligned} &\vdash .\text{sg}ʻ(R_{x}) = (\overrightarrow{R})(x)\\ \text{and}\qquad &\vdash .\text{sg}ʻ{R_{(x,y)}} = (\overrightarrow{R})(x_{y}) \end{aligned} \] as was proved in *65. Hence in particular \[ \vdash .\text{sg}ʻ{\text{ sm }_{(\alpha ,\beta )}} = \text{Nc}(\alpha _{\beta }). \]

It is chiefly for this reason that it is worth while to introduce the definition of \(R(x_{y})\).

We have, in virtue of the above, as will be proved in *102·46, \[ \vdash : \gamma \in tʻ\alpha .\delta \in tʻ\beta .\gamma\,\text{ sm }\,\delta . \equiv .\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \]

With regard to "\(\text{Nc}(\alpha)\)," which is to be interpreted by *65·04, some caution is necessary. This will mean some one of those typically different relations called "\(\text{Nc}\)" which have their domains composed of terms of the same type as \(\alpha\). But it will not mean the logical sum of all such relations, because these relations are of different types according as their converse[Pg 26] domains differ in type, and therefore their logical sum is meaningless. Thus for example if the type of \(\beta\) is lower than or equal to that of \(\alpha\), we shall have \[ \vdash .\exists !\text{Nc}(\alpha )ʻ\beta , \] whence, if "\(\text{Nc}(\alpha)\)" has its converse domain composed of terms of the same type as \(\beta\), \[ \vdash .\Lambda {\sim}\in \text{D}ʻ\text{Nc}(\alpha ). \] But if \(\beta\) is of higher type than \(\alpha\), we shall find \[ \vdash .\Lambda \in \text{D}ʻ\text{Nc}(\alpha). \] Thus "\(\text{Nc}(\alpha)\)" is indeterminate in a way that makes a practical difference.

Exactly similar remarks apply to \(\text{NC}(\alpha)\). We have \[ \vdash .\text{NC}(\alpha ) = \text{D}ʻ\text{Nc}(\alpha ); \] thus "\(\text{NC}(\alpha)\)" shares the ambiguity of "\(\text{Nc}(\alpha)\)." The question whether \(\Lambda \in\text{NC}(\alpha)\) depends upon the decision of this ambiguity. The difficulty is that "\(\text{NC}(\alpha)\)" stands for the domain of any one determination of "\(\text{Nc}\)" which has its domain composed of objects of the type of \(\iota ʻ\alpha\); but it is the domain of only one such determination of "\(\text{Nc}\)," because different determinations are of different types, and therefore cannot be taken together, even when their domains are all of the same type. In consequence of this ambiguity, "\(\text{NC}(\alpha)\)" is a symbol which is as a rule better avoided, and "\(\text{Nc}(\alpha)\)" is not often useful except as a descriptive function, in which case the relatum supplies the requisite typical definiteness.

The peculiarity of "\(\text{NC}(\alpha)\)" is that it is typically definite, and yet is capable of different meanings: it is not wholly definite, being defined as the domain of a relation whose converse domain is typically ambiguous. It results that we cannot profitably make "\(\text{NC}\)" half-definite, as "\(\text{NC}(\alpha)\)" does, but must make it completely definite, as we do by taking \(\text{D}ʻ\text{Nc}(\alpha_{\beta})\). For this we adopt the notation \(\text{NC}^\beta (\alpha)\). We cannot adopt the notation \(\text{NC}(\alpha_{\beta})\), because that would conflict with *65·11, nor \(\text{NC}(\alpha)_{\beta}\), because that would conflict with *65·01, nor \(\text{NC}_{\beta}(\alpha)\), for the same reason. But \(\text{NC}^\beta(\alpha)\) has no previously defined meaning. We may if we like regard "\(\text{NC}^\beta\)" as \(\text{D}ʻ(\text{Nc}\upharpoonright tʻ\beta)\). Then the required meaning of "\(\text{NC}^\beta(\alpha)\)" would result from *65·04. But as "\(\text{NC}^\beta\)" so defined is not required, it is simpler to regard "\(\text{NC}^\beta(\alpha)\)" as a single symbol. We therefore put

*102·01. \(\text{NC}^\beta (\alpha ) = \text{D}ʻ\text{Nc}(\alpha _{\beta }) \quad\text{Df}\)

The present number begins with various propositions (*102·2—·27) on a typically definite relation of similarity, i.e. \(\text{ sm }_{(\alpha,\beta)}\). We then have a set of propositions (*102·3-·46) on "\(\text{Nc}(\alpha_{\beta})ʻ\delta\)." This is only significant if \(\beta\) and \(\delta\) are of the same type; it then denotes the class of those classes which are similar to \(\delta\) and of the same type as \(\alpha\). We then have a set of propositions (*102·5-·64) on \(\text{NC}^\beta(\alpha)\), i.e. on cardinals consisting of classes of the same type as \(\alpha\) which are similar to classes of the same type as \(\beta\). We next prove[Pg 27] (*102·71—·75) that no sub-class of \alpha is similar to \(\text{Cl}ʻ\alpha\), and therefore (substituting \(t_{0}ʻ\alpha\) for \(\alpha\)) no class of the same type as \(\alpha\) is similar to \(tʻ\alpha\), and therefore

*102*74. \(\vdash . \Lambda \in \text{NC}^{tʻ\alpha }(\alpha)\)

This proves that \(\Lambda\) is a cardinal, which is a proposition constantly required. The remaining propositions of *102 are concerned with \(\text{ sm }ʻʻ\mu\) where \(\mu\) is a typically definite cardinal.

The most useful propositions in this number (apart from *102·74) are

*102·3. \(\vdash :\gamma\,\text{ sm }_{(\alpha,\beta)}\delta . \equiv .\gamma \in \text{Nc}(\alpha_{\beta})ʻ\delta\)

*102·46. \(\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\equiv .\delta \in \text{Nc}(\beta _{\alpha })ʻ\gamma . \equiv .\gamma\,\text{ sm }\,\delta .\gamma \in tʻ\alpha .\delta \in tʻ\beta \)

*102·5. \(\vdash :\mu \in \text{NC}^\beta (\alpha ). \equiv .(\exists \delta ).\mu = \text{Nc}(\alpha _{\beta })ʻ\delta \)

*102·6. \(\vdash .\text{Nc}(\alpha )ʻ\beta = \text{Nc}(\alpha _{\beta })ʻ\beta = \hat{\gamma} (\gamma\,\text{ sm }\,\beta .\gamma \in tʻ\alpha ) = \text{Nc}ʻ\beta \cap tʻ\alpha \)

*102·72. \(\vdash :\beta \subset \alpha .\supset .{\sim}(\beta \text{ sm } \text{Cl}ʻ\alpha)\)

This is used in proving \(\mu \in \text{NC}.\supset .2^\mu >\mu\), which is the proposition from which Cantor deduced that there is no greatest cardinal. (If \(\mu = \text{Nc}ʻ\alpha\), \(2^\mu =\text{Nc}ʻ\text{Cl}ʻ\alpha\), and thus there is a rise of type.)

*102·84. \(\vdash :(\exists \gamma ).\gamma \in tʻ\alpha .\gamma \text{ sm } \alpha .\delta \text{ sm } \gamma . \equiv .\delta \text{ sm } \alpha\)

*102·85. \(\vdash .\text{ sm }ʻʻ\mu \cap tʻ\beta = \text{ sm }_{\beta }ʻʻ\mu \)

*102·01. \(\text{NC}^\beta (\alpha ) = \text{D}ʻ\text{Nc}(\alpha _{\beta }) \quad\text{Df}\)

*102·11. \(\vdash :R \in 1\rightarrow 1.\supset .R_{(x,y)} \in 1(x)\rightarrow 1(y)\)

Here, if \(R\) is a real variable, the conditions of significance require \(R = R_{(x,y)}\). But if \(R\) is a typically ambiguous constant, such as \(\iota\) or \(\dot{\Lambda}\) or \(\text{sg}\), \(R_{(x,y)}\) is a typically definite constant. It is chiefly for such cases that propositions such as the above are useful.

Dem. \[ \begin{array}{l} \vdash .*37·402.(*65·1).&\supset \vdash .\text{D}ʻR_{(x,y)}\subset tʻx.\\ [*33·15]&\supset \vdash .\{\text{sg}ʻR_{(x,y)}\}ʻz\subset tʻx.\\ [*63·5]&\supset \vdash .\{\text{sg}ʻR_{(x,y)}\}ʻz \in tʻtʻx &\qquad \text{(1)}\\ \vdash .(1).*71·102.\supset \vdash :\text{Hp}.z \in \text{ᗡ}ʻR_{(x,y)}.&\supset .\{\text{sg}ʻR_{(x,y)}\}ʻz \in 1\cap tʻtʻx.\\ [(*65·02)] &\supset .\{\text{sg}ʻR_{(x,y)}\}ʻz \in 1(x) &\qquad \text{(2)}\\ \text{Similarly}\qquad &\vdash : \text{Hp}.\omega \in \text{D}ʻR_{(x,y)}.\supset .\{gsʻR_{(x,y)}\}ʻ\omega \in 1(y) &\qquad \text{(3)}\\ \vdash .(2).(3).*70·1.\supset \vdash .\text{Prop} \end{array} \]

*102·13. \(\vdash :R \in 1\rightarrow 1.\supset .R_{x} \in 1(x)\rightarrow 1 \quad[\text{Proof as in *102·11}]\)

*102·2. \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )}\delta . \equiv .\gamma \text{ sm } \delta .\gamma \in tʻ\alpha .\delta \in tʻ\beta \quad[*35·102.(*65·1)]\)

*102·21. \[\begin{aligned}\vdash :\gamma \text{ sm }_{(\alpha ,\beta )}&\delta . \equiv .(\exists R).R \in 1\rightarrow 1.\text{D}ʻR \in tʻ\alpha .\\ &\text{ᗡ}ʻR \in tʻ\beta .\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \quad[*102·2.*73·1]\end{aligned}\]

[Pg 28]

*102·22. \(\vdash :\gamma \text{ sm }(x,y)\delta . \equiv .\gamma \text{ sm } \delta .\gamma \subset tʻx.\delta \subset tʻy \quad[*63·5.(*65·12)]\)

102·23. \[\begin{aligned}\vdash :\gamma \text{ sm }(x,y)&\delta . \equiv .(\exists R).R \in 1\rightarrow 1.\text{D}ʻR\subset tʻx.\\ &\text{ᗡ}ʻR\subset tʻy.\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \quad[*102·22.*73·1]\end{aligned}\]

*102·24. \(\vdash :\gamma \text{ sm } (x,y) \delta . \equiv .(\exists R).R \in 1(x)\rightarrow 1(y).\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \)

Dem. \[ \begin{array}{l} \vdash .*102·23.*40·5·52·43.*37·25.\supset \\ \vdash \colon\ldotp \gamma \text{ sm }v(x,y)\delta . &\equiv :(\exists R):R \in 1\rightarrow 1 : w \in \text{ᗡ}ʻR.\supset _{w}\overrightarrow{R}ʻw\subset tʻx:\\ &z \in \text{D}ʻR.\supset _{z}.\overleftarrow{R}ʻz\subset tʻy : \text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta :\\ [*63·5] &\equiv :(\exists R):R \in 1\rightarrow 1 . \overleftarrow{R}ʻʻ\text{ᗡ}ʻR\subset tʻtʻx.\overleftarrow{R}ʻʻ\text{D}ʻR\subset tʻtʻy.\\ &\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta :\\ [*71·102.(*65·02)] &\equiv :(\exists R).\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset 1(x).\overleftarrow{R}ʻʻ\text{D}ʻR\subset 1(y).\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta :\\ [*70·1] &\equiv :(\exists R).R \in 1(x)\rightarrow 1(y).\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*102·25. \[\begin{aligned}&\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta . \equiv .(\exists R).R \in 1_{\alpha }\rightarrow 1_{\beta }.\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta\\ &[\text{Proof as in *102·24}]\end{aligned}\]

*102·26. \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta .\gamma'\text{ sm }_{(\alpha ,\beta )} \delta .\supset .\gamma \text{ sm }_{(\alpha ,\alpha )}\gamma'\)

Dem. \[ \begin{array}{l} \vdash .*102·2. \supset \vdash : \text{Hp}.&\supset .\gamma \text{ sm } \delta .\gamma'\text{ sm } \delta .\gamma ,\gamma'\in \tau ʻ\alpha .\\ [*73·32] &\supset .\gamma \text{ sm } \gamma'.\gamma ,\gamma'\in \tau ʻ\alpha .\\ [*102·2] &\supset .\gamma \text{ sm }_{(\alpha ,\alpha )}\gamma':\supset \vdash .\text{Prop} \end{array} \]

*102·27. \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta .\gamma' \text{ sm }_{(\alpha ʻ,\beta )} \delta .\supset .\gamma \text{ sm }_{(\alpha ,\alpha ʻ)}\gamma' \quad[\text{Proof as in *102·26}]\)

*102·3. \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta . \equiv .\gamma \in \text{Nc}{(\alpha _\beta )}ʻ\delta \)

Dem. \[ \begin{array}{l} \vdash .*32·18.\supset \\ \vdash :\gamma \text{ sm }_{(\alpha _\beta )} \delta . &\equiv .\gamma \in {\text{sg}ʻ\text{ sm }_(\alpha _{\beta })}ʻ\delta .\\ [*65·2] &\equiv .\gamma \in {(\text{sg}ʻ\text{ sm })(\alpha _{\beta })}ʻ\delta .\\ [(*100·01)] &\equiv \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta :\supset \vdash .\text{Prop} \end{array} \]

*102·31. \(\vdash .\text{Nc} (\alpha _{\beta })ʻ\delta = \text{D}ʻʻ\{1\rightarrow 1\cap \hat{R} (\text{D}ʻR \in tʻ\alpha .\text{ᗡ}ʻR \in tʻ\beta .\text{ᗡ}ʻR = \delta )\}\)

Dem. \[ \begin{array}{l} \vdash .*102·3·21.\supset\\ \vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . &\equiv .(\exists R).R \in 1\rightarrow 1.\text{D}ʻR \in tʻ\alpha .\text{ᗡ}ʻR \in tʻ\beta .\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta .\\ [*33·123.*37·1] &\equiv .\gamma \in \text{D}ʻʻ\{1\rightarrow 1\cap \hat{R} (\text{D}ʻR \in tʻ\alpha . \text{ᗡ}ʻR \in tʻ\beta .\text{ᗡ}ʻR = \delta )\}:\\ \supset \vdash .\text{Prop} \end{array} \]

[Pg 29]

*102·32. \(\vdash . \text{Nc}(\alpha _{\beta })ʻ\delta = \text{D}ʻʻ\{(1_{\alpha }\rightarrow 1_{\beta }) \cap \overleftarrow{\text{ᗡ}}ʻ\delta\}\)

Dem. \[ \begin{array}{l} \vdash .*102·3·25. \supset \\ \vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . &\equiv . (\exists R) . R \in 1_{\alpha }\rightarrow 1_{\beta } . \text{D}ʻR = \gamma . \text{ᗡ}ʻR = \delta .\\ [*33·61] &\equiv . (\exists R) . R \in 1_{\alpha }\rightarrow 1_{\beta } . R \in \overleftarrow{\text{ᗡ}}ʻ\delta . \text{D}ʻR = \gamma .\\ [*33·123.*37·1] &\equiv . \gamma \in \text{D}ʻʻ{(1_{\alpha }\rightarrow 1_{\beta }) \cap \overleftarrow{\text{ᗡ}}ʻ\delta } : \supset \vdash . \text{Prop} \end{array} \]

*102·34. \[\begin{aligned}\vdash . \text{Nc}(\alpha ,\beta )ʻ\delta = &\text{D}ʻʻ\{1 \rightarrow 1 \cap \hat{R} (\text{D}ʻR \in tʻ\alpha . \text{ᗡ}ʻR \subset tʻ\beta . \text{ᗡ}ʻR = \delta )\}\\ &[\text{Proof as in *102·31}]\end{aligned}\]

*102·35. \(\vdash . \text{Nc}(\alpha ,\beta )ʻ\delta = \text{D}ʻʻ[{1_{\alpha } \rightarrow 1(\beta )} \cap \overleftarrow{\text{ᗡ}}ʻ\delta ] \quad[\text{Proof as in *102·32}]\)

*102·36. \(\vdash . \text{E}! \text{Nc}(\alpha _{\beta })ʻ\delta \quad[*102·31.*14·21]\)

This proposition is true whenever it is significant, and is significant whenever \(\delta \in tʻ\beta\). When \(\delta\) belongs to some other type, the above proposition is not significant.

*102·361. \(\vdash . \text{E}! \text{Nc}(\alpha , \beta )ʻ\delta \quad[*102·34.*14·21]\)

*102·37. \(\vdash . \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) = tʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*37·402.(*65·11). &\supset \vdash . \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) \subset tʻ\beta &&\qquad \text{(1)}\\ \vdash .*102·36.*33·43 . &\supset \vdash . (\delta ) . \delta \in \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }). [*63·14] &\supset \vdash . t_{0}ʻ\text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) = \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) &\qquad \text{(2)}\\ \vdash .(1).*63·21. &\supset \vdash .t_{0}ʻ\text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) = tʻ\beta &&\qquad \text{(3)}\\ \vdash .(2).(3). \supset \vdash . \text{Prop} \end{array} \]

*102·4. \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \gamma' \in \text{Nc}(\alpha _{\beta })ʻ\delta . \supset . \gamma \in \text{Nc}(\alpha _{\alpha })ʻ\gamma'\quad[*102·3·26]\)

*102·41. \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \gamma'\in \text{Nc}(\alpha _{\beta }ʻ)ʻ\delta . \supset . \gamma \in \text{Nc}(\alpha _{\alpha ʻ})ʻ\gamma' \quad[*102·3·27]\)

*102·42. \(\vdash . \alpha \in \text{Nc}(\alpha _{\alpha })ʻ\alpha \quad[*102·3·2.*73·3.*63·103]\)

*102·43. \(\vdash . \exists ! \text{Nc}(\alpha _{\alpha })ʻ\alpha \quad[*102·42]\)

This inference is legitimate because, when \(\alpha\) is given, "\(\text{Nc}(\alpha _{\alpha })ʻ\alpha\)" is typically definite. The inference from "\(\alpha \in \text{Nc}ʻ\alpha\)" (which is true) to "\(\exists !\text{Nc}ʻ\alpha\)" is not valid, because "\(\exists!\text{Nc}ʻ\alpha\)" may hold only for some of the possible determinations of the ambiguity of "\(\text{Nc}\)."

*102·44. \(\vdash : \alpha \text{ sm } \beta . \equiv . \alpha \in \text{Nc}(\alpha _{\beta })ʻ\beta . \equiv . \beta \in \text{Nc}(\beta _{\alpha })ʻ\alpha\)

Dem. \[ \begin{array}{l} &\vdash .*63·102 . \supset\\ &\vdash : \alpha \text{ sm } \beta . \equiv . \alpha \text{ sm } \beta . \alpha \in tʻ\alpha . \beta \in tʻ\beta &\qquad \text{(1)}\\ &\vdash .(1).*102·2·3 . \supset \vdash . \text{Prop} \end{array} \]

[Pg 30]

*102·45. \(\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\supset .\gamma \in \text{Nc}(\alpha _{\alpha })ʻ\gamma \)

Dem. \[ \begin{array}{l} &\vdash .*102·3·2. \supset \vdash :\text{Hp}.\supset .\gamma \in tʻ\alpha &\qquad \text{(1)}\\ &\vdash .*73·3. \supset \vdash .\gamma \text{ sm } \gamma &\qquad \text{(2)}\\ &\vdash .(1).(2).*102·3·2.\supset \vdash .\text{Prop} \end{array} \]

*102·46. \[\begin{aligned}&\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \equiv .\delta \in \text{Nc}(\beta _{\alpha }ʻ\gamma . \equiv .\gamma \text{ sm } \delta .\gamma \in tʻ\alpha.\delta \in tʻ\beta \\ &[*102·2·3.*73·31]\end{aligned}\]

*102·5. \(\vdash :\mu \in \text{NC}^\beta (\alpha ). \colon\ldotp .(\exists \delta ).\mu = \text{Nc}(\alpha _{\beta })ʻ\delta \quad[*100·22.*71·41.(*102*01)]\)

In using propositions, such as those of *100, in which we have a typically ambiguous "\(\text{Nc}\)" or "\(\text{NC}\)," any significant typical definiteness may be added, since, when a typically ambiguous proposition is asserted, that includes the assertion of every possible proposition resulting from determining the ambiguity.

*102·501. \(\vdash .\text{Nc}(\alpha _{\beta }ʻ\delta \in \text{NC}^\beta (\alpha ) \quad[*102·5·36]\)

*102·51. \[\begin{aligned}\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\supset .&\text{Nc}((\alpha _{\beta })ʻ\delta = \text{Nc}(\alpha _{\alpha })ʻ\gamma .\\ &\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{NC}^\beta (\alpha ).\text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\alpha (\alpha )\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*102·3·2.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\gamma \text{ sm } \delta .\gamma \in tʻ\alpha .\delta \in tʻ\beta :\\ [*73·37.*4·73] \supset :\xi \text{ sm } \delta . \equiv .&\xi \text{ sm } \gamma :\xi \text{ sm } \delta . \equiv .\xi \text{ sm } \delta .\delta \in tʻ\beta :\\ &\xi \text{ sm } \gamma . \equiv .\xi \text{ sm } \gamma .\gamma \in tʻ\alpha :\\ [*4·22] &\supset :\xi \text{ sm } \delta .\delta \in tʻ\beta . \equiv .\xi \text{ sm } \gamma .\gamma \in tʻ\alpha :\\ [\text{Fact}] &\supset :\xi \text{ sm } \delta .\xi \in tʻ\alpha .\delta \in tʻ\beta . \equiv .\xi \text{ sm } \gamma .\xi \in tʻ\alpha .\gamma \in tʻ\alpha :\\ [*102·2·3] &\supset :\text{Nc}(\alpha _{\beta })ʻ\delta = \text{Nc}(\alpha _{\alpha })ʻ\gamma &\qquad \text{(1)}\\ \vdash .(1).*102·501 .\supset \vdash .\text{Prop} \end{array} \]

*102·52. \(\vdash :\exists ! \text{Nc}(\alpha _{\beta })ʻ\delta .\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{NC}^\alpha (\alpha ) \quad[*102·51]\)

*102·53. \(\vdash .\text{NC}^\beta (\alpha )-\iota ʻ\Lambda \subset \text{NC}^\alpha (\alpha )\)

Dem. \[ \begin{array}{l} \vdash .*102·52.\supset \vdash :\mu = \text{Nc}(\alpha _{\beta })ʻ\delta .\exists !\mu .\supset .\mu \in \text{NC}^\alpha (\alpha ) &\qquad \text{(1)}\\ \vdash .(1).*102*5.\supset \vdash .\text{Prop} \end{array} \]

*102·54. \(\vdash :\delta \in \text{Nc}(\beta _{\alpha })ʻ\gamma .\supset .\text{Nc}(\alpha _{\beta })ʻ\delta = \text{Nc}(\alpha _{\alpha })ʻ\gamma \quad[*102·51·46]\)

*102·541. \(\vdash :\exists ! \text{Nc}(\beta _{\alpha })ʻ\gamma .\supset .\text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha )-tʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash . *102·54·501.&\supset \vdash :\delta \in \text{Nc}(\beta_{\alpha}ʻ\gamma .\supset .\text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha ) &\qquad \text{(1)}\\ \vdash . *102·46·45. \supset :\vdash \delta \in \text{Nc}{(\beta_{\alpha})}ʻ\gamma .&\supset .\gamma \in \text{Nc}(\alpha _{\alpha })ʻ\gamma .\\ [*10·24] &\supset .\exists !\text{Nc}(\alpha_{\alpha})ʻ\gamma &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \\ \vdash :\delta \in \text{Nc}(\beta_{\alpha})ʻ\gamma .&\supset .\text{Nc}(\alpha_{\alpha})ʻ\gamma \in \text{NC}^\beta (\alpha )-tʻ\Lambda :\supset \vdash .\text{Prop} \end{array} \]

[Pg 31]

*102·55. \(\vdash : \Lambda {\sim}\in \text{NC}^\alpha (\beta ) . \supset . \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda = \text{NC}^\alpha (\alpha )\)

Dem. \[ \begin{array}{l} \vdash .*102·5 . \supset \\ \vdash \colon\ldotp \text{Hp} . &\supset : \mu = \text{Nc}(\beta _{\alpha })ʻ\gamma . \supset _{\mu ,\gamma } . \exists !\mu .\\ [*102·541] &\supset _{\mu ,\gamma } . \text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ [*10·23] &\supset : (\exists \mu ) . \mu = \text{Nc}(\beta _{\alpha })ʻ\gamma . \supset _{\gamma } . \text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ [*102·36] &\supset : (\gamma ) . \text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ [*13·191] &\supset : \nu = \text{Nc}(\alpha _{\alpha })ʻ\gamma . \supset _{\nu ,\gamma } . \nu \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ [*102·5] &\supset : \nu \in \text{NC}^\alpha (\alpha ) . \supset _{\nu } . \nu \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash .(1).*102·53. \supset \vdash . \text{Prop} \end{array} \]

The above proposition shows that, if every class of the same type as \(\beta\) is similar to some class of the same type as \(\alpha\), then, given a class \(\gamma\) of the same type as \(\alpha\), there is a class \(\delta\), of the same type as \(\beta\), such that the classes similar to \(\delta\) and of the same type as \(\alpha\) are the same as the classes similar to \(\gamma\) and of the same type as \(\alpha\); and conversely, given any class \(\delta\), of the same type as \(\beta\), and similar to some class of the same type as \(\alpha\), then there is a class \(\gamma\), of the same type as \(\alpha\), such that the classes similar to \(\gamma\) and of the same type as \(\alpha\) are the same as the classes similar to \(\delta\) and of the same type as \(\alpha\). We may express this by saying that, if the cardinals which go from the type of \(\alpha\) to the type of \(\beta\) are never null, then those that go from the type of \(\beta\) to the type of \(\alpha\), with the exception of \(\Lambda\) (if \(\Lambda\) is one of them), are the same as those that begin and end within the type of \(\alpha\). The latter are what we call "homogeneous" cardinals. Thus our proposition is a step towards reducing the general study of cardinals to that of homogeneous cardinals.

*102·6. \(\vdash . \text{Nc}(\alpha )ʻ\beta = \text{Nc}(\alpha _{\beta })ʻ\beta = \hat{\gamma} (\gamma \text{ sm } \beta . \gamma \in tʻ\alpha ) = \text{Nc}ʻ\beta \cap tʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*35·1.(*65·04). \supset \\ \vdash : \mu = \text{Nc}(\alpha )ʻ\beta . &\equiv . \mu = \text{Nc}ʻ\beta . \mu \in t^{2}ʻ\alpha .\\ [*63·5] &\equiv . \mu = \text{Nc}ʻ\beta . \mu \subset tʻ\alpha .\\ [*65·13] &\equiv . \mu = \text{Nc}ʻ\beta \cap tʻ\alpha . &&\qquad \text{(1)}\\ [*100·1] &\equiv . \mu = \hat{\gamma} (\gamma \text{ sm } \beta . \gamma \in tʻ\alpha ). &&\qquad \text{(2)}\\ [*63·103] &\equiv . \mu = \hat{\gamma} (\gamma \text{ sm } \beta . \gamma \in tʻ\alpha . \beta \in tʻ\beta ). [*102·46] &\equiv . \mu = \text{Nc}(\alpha _{\beta })ʻ\beta &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*20·2.*100·1. \supset \vdash . \text{Prop} \end{array} \]

*102·61. \(\vdash : \delta \in tʻ\beta . \supset . \text{Nc}(\alpha )ʻ\delta = \text{Nc}(\alpha _{\beta })ʻ\delta \)

Dem. \[ \begin{array}{l} \vdash .*4·73 . \supset \vdash : \text{Hp} . \supset . \hat{\gamma} (\gamma \text{ sm } \delta . \gamma \in tʻ\alpha ) &= \hat{\gamma} (\gamma \text{ sm } \delta . \gamma \in tʻ\alpha . \delta \in tʻ\beta )\\ [*102·46] &= \text{Nc}(\alpha _{\beta })ʻ\delta &\qquad \text{(1)}\\ \vdash .(1).*102·6. \supset \vdash . \text{Prop} \end{array} \]

[Pg 32]

*102·62. \(\vdash . \text{NC}^\beta (\alpha ) = \text{Nc}(\alpha )ʻʻtʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*37·7.(*100·01). \supset \\ \vdash . \text{Nc}(\alpha )ʻʻtʻ\beta &= \hat{\mu} \{(\exists \delta ) . \delta \in tʻ\beta . \mu = \text{Nc}(\alpha )ʻ\delta\}\\ [*102·61] &= \hat{\mu} \{(\exists \delta ) . \delta \in tʻ\beta . \mu = \text{Nc}(\alpha _{\beta})ʻ\delta\}\\ [*102·37] &= \text{D}ʻ\text{Nc}(\alpha _{\beta })\\ [(*102·01)] &= \text{NC}^\beta (\alpha ) . \supset \vdash . \text{Prop} \end{array} \]

*102·63. \(\vdash : \mu = \text{Nc}ʻ\gamma . \alpha \in \mu . \supset . \mu = \text{Nc}(\alpha )ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *63·5 . \supset \vdash : \text{Hp} . &\supset . \mu = \text{Nc}ʻ\gamma . \mu \subset tʻ\alpha .\\ [*65·13] &\supset . \mu = \text{Nc}ʻ\gamma \cap tʻ\alpha .\\ [*102·6] &\supset . \mu = \text{Nc}(\alpha )ʻ\gamma : \supset \vdash . \text{Prop} \end{array} \]

*102·64. \(\vdash : \mu \in \text{NC} . \exists !\mu . \supset . (\exists \alpha ,\gamma ) . \mu = \text{Nc}(\alpha )ʻ\gamma \quad[*102·63.*100·4]\)

The following propositions are part of Cantor's proof that there is no greatest cardinal. They are inserted here in order to enable us to prove that \(\Lambda\) is a cardinal, namely what we call a "descending" cardinal, i.e. one whose corresponding "\(\text{ sm }\)" goes from a higher to a lower type.

*102·71. \(\vdash : R \in \text{Cls} \rightarrow 1 . \text{D}ʻR\subset \alpha . \text{ᗡ}ʻR\subset \text{Cl}ʻ\alpha . \supset . \exists ! \text{Cl}ʻ\alpha - \text{ᗡ}ʻR\)

Dem. \[ \begin{array}{l} \vdash .*20·33.*4·73 . \supset \\ \vdash \colon\colon \text{Hp} . \varpi = &\hat{x} (x \in \text{D}ʻR . x {\sim}\in \breve{R}ʻx) . \supset \colon\ldotp \\ x \in \text{D}ʻR . &\supset _{x} : x \in \varpi . \equiv . x {\sim} \in \breve{R}ʻx:\\ [*5·18] &\supset _{x} : {\sim}\{x \in \varpi . \equiv . x \in \breve{R}ʻx\}:\\ [*20·43.\text{Transp}.*71·164] &\supset _{x} : \varpi \neq \breve{R}ʻx\colon\ldotp \\ [*71·411.\text{Transp}] &\supset \colon\ldotp \varpi {\sim}\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ \vdash .*20·33.*3·26 . \supset \vdash : \text{Hp}(1) . &\supset . \varpi \subset \text{D}ʻR.\\ [\text{Hp}] &\supset . \varpi \subset \alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*13·191 . \supset \\ \vdash : \text{Hp} . \supset . \hat{x} (x \in \text{D}ʻR . x {\sim}\in \breve{R}ʻx) \in \text{Cl}ʻ\alpha - \text{ᗡ}ʻR : \supset \vdash . \text{Prop} \end{array} \]

*102·72. \(\vdash : \beta \subset \alpha . \supset .{\sim}(\beta \text{ sm } \text{Cl}ʻ\alpha )\)

Dem. \[ \begin{array}{l} \vdash .*102·71. \supset \vdash \colon\ldotp \text{Hp} . &\supset : R \in 1 \rightarrow 1 . \text{D}ʻR = \beta . \text{ᗡ}ʻR \subset \text{Cl}ʻ\alpha . \supset _{R} . \exists ! \text{Cl}ʻ\alpha - \text{ᗡ}ʻR:\\ [*24·55.*22·41] &\supset : R \in 1 \rightarrow 1 . \text{D}ʻR = \beta . \supset _{R}. \text{ᗡ}ʻR \neq \text{Cl}ʻ\alpha :\\ [*10·51] &\supset : {\sim}(\exists R) . R \in 1 \rightarrow 1 . \text{D}ʻR = \beta . \text{ᗡ}ʻR = \text{Cl}ʻ\alpha :\\ [*73·1] &\supset : {\sim}(\beta \text{ sm } \text{Cl}ʻ\alpha ) \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

[Pg 33]

*102·73. \(\vdash . \text{Nc}(\alpha )ʻtʻ\alpha = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*102·6 . \supset \vdash . \text{Nc}(\alpha )ʻtʻ\alpha &= \hat{\gamma} (\gamma \text{ sm } tʻ\alpha . \gamma \in tʻ\alpha )\\ [*63·65] &= \hat{\gamma} (\gamma \text{ sm } \text{Cl}ʻt_{0}ʻ\alpha . \gamma \subset t_{0}ʻ\alpha )\\ [*102·72] &= \Lambda . \supset \vdash . \text{Prop} \end{array} \]

This proposition proves that no class of the same type as \(\alpha\) is similar to \(tʻ\alpha\). Now \(tʻ\alpha\) is the greatest class of its type; thus there are classes of the type next above that of \(\alpha\) which are too great to be similar to any class of the type of \(\alpha\). Thus (as will be explicitly proved later) the maximum cardinal in one type is less than that in the next higher type. Cantor's proposition that there is no maximum cardinal only holds when we are allowed to rise to continually higher types: in each type, there is a maximum for that type, namely the number of members of the type.

*102·74. \(\vdash . \Lambda \in \text{NC}^{tʻ\alpha }(\alpha )\)

Dem. \[ \begin{array}{l} \vdash .*102·6·501. &\supset \vdash . \text{Nc}(\alpha )ʻtʻ\alpha \in \text{NC}^{tʻ\alpha }(\alpha ) &\qquad \text{(1)}\\ \vdash .(1).*102·73. &\supset \vdash .\text{Prop} \end{array} \]

*102·75. \(\vdash . \text{NC}^{tʻ\alpha }(\alpha ) = \text{NC}^\alpha (\alpha ) \cup \iota ʻ\Lambda \)

Dem. \[ \begin{array}{l} \vdash .*100·6. &\supset \vdash : \gamma \in tʻ\alpha . \supset . \gamma \in tʻ\alpha . \iota ʻʻ\gamma \in \text{Nc}ʻ\gamma &\qquad \text{(1)}\\ \vdash .*63·64·5. &\supset \vdash : \gamma \in tʻ\alpha . \supset . \iota ʻʻ\gamma \in t^{2}ʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*102·46. \supset \vdash : \gamma \in tʻ\alpha . &\supset . \iota ʻʻ\alpha \in \text{Nc}{(tʻ\alpha )_{\alpha }}ʻ\gamma .\\ [*10·24] &\supset . \exists ! \text{Nc}{(tʻ\alpha )_{\alpha }}ʻ\gamma &\qquad \text{(3)}\\ \vdash .(3).*102·55. &\supset \vdash . \text{NC}^{tʻ\alpha }(\alpha ) - \iota ʻ\Lambda = \text{NC}^\alpha (\alpha ) &\qquad \text{(4)}\\ \vdash .(4).*102·74. &\supset \vdash .\text{Prop} \end{array} \]

*102·8. \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \gamma \text{ sm } \zeta . \zeta \in tʻ\xi . \supset . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *102·46 . \supset \\ \vdash : \text{Hp}. &\supset . \gamma \text{ sm } \delta . \gamma \in tʻ\alpha . \delta \in tʻ\beta . \gamma \text{ sm } \zeta . \zeta \in tʻ\xi .\\ [*73·31·32] &\supset . \zeta \text{ sm } \delta . \zeta \in tʻ\xi . \delta \in tʻ\beta . \zeta \text{ sm } \gamma . \zeta \in tʻ\xi . \gamma \in tʻ\alpha .\\ [*102·46] &\supset . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma : \supset \vdash . \text{Prop} \end{array} \]

*102·81. \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \supset . \text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi = \text{Nc}(\xi _{\beta })ʻ\delta = \text{Nc}(\xi _{\alpha })ʻ\gamma \)

Dem. \[ \begin{array}{l} \vdash .*102·8.*73·31·32 . \supset \\ \vdash : \gamma , \gamma ʻ &\in \text{Nc}(\alpha _{\beta })ʻ\delta . \zeta \text{ sm } \gamma ʻ . \zeta \in tʻ\xi . \supset . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma &\qquad \text{(1)}\\ \vdash .(1).*37·1. \supset \vdash : \text{Hp} . \supset . &\text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi \subset \text{Nc}(\xi _{\beta })ʻ\delta .\\ &\text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi \subset \text{Nc}(\xi _{\alpha })ʻ\gamma &\qquad \text{(2)}\\ \vdash .*102·46.*73·31·32 . \supset \\ \vdash : \text{Hp} . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta . &\supset . \zeta \in tʻ\xi . \zeta \text{ sm } \gamma . \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\\ [*37·1] &\supset . \zeta \in \text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi &\qquad \text{(3)}\\ \text{Similarly}\\ \vdash : \text{Hp} . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma . &\supset . \zeta \in \text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi &\qquad \text{(4)}\\ \vdash .(2).(3).(4). \supset \vdash . \text{Prop} \end{array} \]

[Pg 34]

*102·82. \(\vdash : \mu \in \text{NC}^\beta (\alpha ) . \exists !\mu . \supset . \text{ sm }ʻʻ\mu \cap tʻ\xi \in \text{NC}^\beta (\xi ) \quad[*102·81·5]\)

*102·83. \[\begin{aligned}\vdash : \mu \in \text{NC}^\beta (\alpha ) . &\exists !\mu . \nu = \text{ sm }ʻʻ\mu \cap tʻ\xi . \exists !\nu . \supset .\\ &\text{ sm }ʻʻ\mu \cap tʻ\zeta = \text{ sm }ʻʻ\nu \cap tʻ\zeta . \mu = \text{ sm }ʻʻ\nu \cap tʻ\alpha\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *102·81 . \supset \\ \vdash : \text{Hp} . \gamma \in \mu . &\gamma' \in \nu . \mu = \text{Nc}(\alpha _{\beta })ʻ\delta . \supset .\\ &\nu = \text{Nc}(\xi _{\beta })ʻ\delta = \text{Nc}(\xi _{\alpha })ʻ\gamma . \text{ sm }ʻʻ\mu \cap tʻ\zeta = \text{Nc}(\zeta _{\alpha })ʻ\gamma .\\ [*102·81] \supset . &\text{ sm }ʻʻ\nu \cap tʻ\zeta = \text{Nc} (\zeta _{\alpha })ʻ\gamma = \text{ sm }ʻʻ\mu \cap tʻ\zeta .\\ &\text{ sm }ʻʻ\nu \cap tʻ\alpha = \text{Nc}(\alpha _{\beta })ʻ\delta = \mu &\qquad \text{(1)}\\ \vdash . (1) . *102·5 . \supset \vdash . \text{Prop} \end{array} \]

*102·84. \(\vdash : (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma . \equiv . \delta \text{ sm } \alpha\)

Dem. \[ \begin{array}{l} \vdash . *73·32 . &\supset \vdash : (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma . \supset . \delta \text{ sm } \alpha &\qquad \text{(1)}\\ \vdash . *73·3 . *63·103 . \supset \\ \vdash : \delta \text{ sm } \alpha . &\supset . \alpha \text{ sm } \alpha . \alpha \in tʻ\alpha . \delta \text{ sm } \alpha .\\ [*10·24] &\supset . (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma &\qquad \text{(2)}\\ \vdash . (1) . (2) . \supset \vdash . \text{Prop} \end{array} \]

*102·85. \(\vdash . \text{ sm }ʻʻ\mu \cap tʻ\beta = \text{ sm }_{\beta }ʻʻ\mu \quad[*65·3]\)

*102·86. \(\vdash : \mu = \text{Nc} (\alpha )ʻ\delta . \exists !\mu . \supset . \text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta\)

Dem. \[ \begin{array}{l} \vdash . *102·6·81 . \supset \\ \vdash : \gamma \in \text{Nc}(\alpha )ʻ\delta . &\supset . \text{ sm }ʻʻ\text{Nc}(\alpha )ʻ\delta \cap tʻ\xi = \text{Nc}(\xi )ʻ\delta .\\ [*102·85] &\supset . \text{ sm }_{\xi }ʻʻ\text{Nc}(\alpha )ʻ\delta = \text{Nc}(\xi )ʻ\delta &\qquad \text{(1)}\\ \vdash . (1) . *13·12 . \supset \\ \vdash : \mu = \text{Nc} (\alpha )ʻ\delta . \gamma \in \mu . &\supset . \text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta : \supset \vdash . \text{Prop} \end{array} \]

*102·861. \(\vdash . \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ \mu \subset \text{ sm }_{\alpha }ʻʻ\mu \)

Dem. \[ \begin{array}{l} \vdash . *37·1 . \supset \vdash : \gamma \in \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu . &\supset . (\exists \zeta ,\eta ) . \eta \in \mu . \zeta \text{ sm } \eta . \zeta \in tʻ\xi . \gamma \text{ sm } \zeta . \gamma \in tʻ\alpha .\\ [*73·32.*10·5] &\supset . (\exists \eta ) . \eta \in \mu . \gamma \text{ sm } \eta . \gamma \in tʻ\alpha .\\ [*37·1] &\supset . \gamma \in \text{ sm }_{\alpha }ʻʻ\mu : \supset \vdash . \text{Prop} \end{array} \]

*102·862. \(\vdash \colon\ldotp \eta \in \mu . \supset _{\eta } . \exists !\text{Nc}(\xi )ʻ\eta : \supset . \text{ sm }_{\alpha }ʻʻ\mu = \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu \)

Dem. \[ \begin{array}{l} \vdash . *102·6. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \eta \in \mu . \supset . (\exists \zeta ) . \zeta \text{ sm } \eta . \zeta \in tʻ\xi :\\ [\text{Fact}.*10·35] &\supset : \eta \in \mu . \gamma \text{ sm } \eta . \gamma \in tʻ\alpha . \supset .\\ &(\exists \zeta ) . \zeta \text{ sm } \eta . \zeta \in tʻ \xi . \gamma \text{ sm } \eta . \gamma \in tʻ\alpha .\\ [*73·37] &\supset . (\exists \zeta ) . \zeta \text{ sm } \eta . \zeta \in tʻ \xi . \gamma \text{ sm } \zeta . \zeta \in tʻ\alpha .\\ [*37·1] &\supset . \gamma \in \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(1)}\\ \vdash . (1) . *10·11·23 . *37·1 . &\supset \vdash : \text{Hp}. \supset . \text{ sm }_{\alpha }ʻʻ\mu \subset \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ \vdash . (2) . *102·861 . \supset \vdash . \text{Prop} \end{array} \]

[Pg 35]

*102·863. \(\vdash \colon\ldotp \mu = \text{Nc}(\beta )ʻ\delta . \exists !\text{Nc}(\xi )ʻ\delta . \supset : \eta \in \mu . \supset _{\eta } . \exists !\text{Nc}(\xi )ʻ\eta \)

Dem. \[ \begin{array}{l} \vdash . *100·31·321. \supset \vdash : \text{Hp} . \eta \in \mu . &\supset . \text{Nc}(\xi )ʻ\eta = \text{Nc}(\xi )ʻ\delta .\\ [\text{Hp}] &\supset . \exists !\text{Nc}(\xi )ʻ\eta : \supset \vdash . \text{Prop} \end{array} \]

*102·87. \[\begin{aligned}&\vdash : \mu = \text{Nc}(\beta )ʻ\delta . \exists !\text{Nc}(\xi )ʻ\delta . \supset . \text{ sm }_{\alpha }ʻʻ\mu = \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu \\ &[*102·862·863]\end{aligned}\]

*102·88. \[\begin{aligned}\vdash : \mu = \text{Nc}(\beta )ʻ\delta . \exists !\text{ sm }_{\xi }ʻʻ\mu . \supset . &\text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta . \text{ sm }_{\alpha }ʻʻ\mu = \text{Nc}(\alpha )ʻ\delta .\\ &\text{ sm }_{\alpha }ʻʻ\mu = \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu = \text{ sm }_{\alpha }ʻʻ\text{Nc}(\xi )ʻ\delta\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *37·29 . \text{Transp}. \supset \vdash : \text{Hp} . \supset . \exists !\mu .\\ [*102·86] &\supset . \text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta . \text{ sm }_{\alpha }ʻʻ\mu = \text{Nc}(\alpha )ʻ\delta . &\qquad \text{(1)}\\ [\text{Hp}] &\supset . \exists !\text{Nc}(\xi )ʻ\delta .\\ [*102·87] \supset . \text{ sm }_{\alpha }ʻʻ\mu &= \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ [(1)] &= \text{ sm }_{\alpha }ʻʻ\text{Nc}(\xi )ʻ\delta &\qquad \text{(3)}\\ \vdash . (1) . (2) . (3) . \supset \vdash . \text{Prop} \end{array} \]

Note on negative statements concerning types. Statements such as "\(x {\sim} \in tʻy\)" or "\(x {\sim} \in t_{0}ʻ\alpha\)" are always false when they are significant. Hence when an object belongs to one type, there is no significant way of expressing what we mean when we say that it does not belong to some other type. The reason is that, when, for example, \(tʻ\alpha\) and \( t_{0}ʻ\alpha\) are said to be different, the statement is only significant if interpreted as applying to the symbols, i.e. as meaning to deny that the two symbols denote the same class. We cannot assert that they denote different classes, since "\(tʻ\alpha \neq t_{0}ʻ\alpha\)" is not significant, but we can deny that they denote the same class. Owing to this peculiarity, propositions dealing with types acquire their importance largely from the fact that they can be interpreted as dealing with the symbols rather than directly with the objects denoted by the symbols. Another reason for the importance of typically definite propositions is that, when they are implications of which the hypothesis can be asserted, they can be used for inference, i.e. for the assertion of the conclusion. Where typically ambiguous symbols occur in implications, on the contrary, the conditions of significance may be different for the hypothesis and the conclusion, so that fallacies may arise from the use of such implications in inference. E.g. it is a fallacy to infer "\(\vdash . \exists !\text{Nc}ʻ\alpha\)" from the (true) propositions "\(\vdash : \alpha \in \text{Nc}ʻ\alpha .\supset . \exists !\text{Nc}ʻ\alpha\)" and "\(\vdash . \alpha \in \text{Nc}ʻ\alpha\)." (The truth of the first of these two requires that "\(\text{Nc}ʻ\alpha\)" should receive the same typical determination in both its occurrences.) For these two reasons hypothetical concerning types are often useful, in spite of the fact that their hypotheses are always true when they are significant.

Summary of *103.

In this number, we shall consider cardinals generated by a homogeneous relation of similarity. A "homogeneous" cardinal is to mean all the classes similar to some class \(\alpha\) and of the same type as \(\alpha\). The "homogeneous cardinal of \(\alpha\)" will be defined as \(\text{Nc}ʻ\alpha \cap tʻ\alpha\); we shall denote it by "\(\text{N}_{0}\text{c}ʻ\alpha\)." Then the class of homogeneous cardinals is the class of all such cardinals as "\(\text{N}_{0}\text{c}ʻ\alpha\)," i.e. it is \(\text{D}ʻ\text{N}_{0}\text{c}\); this we shall denote by "\(\text{N}_{0}\text{C}\)." The symbol "\(\text{N}_{0}\text{c}ʻ\alpha\)" is typically definite as soon as \(\alpha\) is assigned; "\(\text{N}_{0}\text{C}\)," on the contrary, is typically ambiguous: it must be a \(\text{Cls}^{3}\), but otherwise its type may vary indefinitely. Homogeneous cardinals have, however, many properties which do not require that the ambiguity of "\(\text{N}_{0}\text{C}\)" should be determined, and few which do require this. They are important also as being the simplest kind of cardinals, and as being a kind to which other kinds can usually be reduced.

The chief advantage of homogeneous cardinals is that they are never null (*103·13·22). This enables us to avoid by their means the explicit exclusion of exceptional cases; thus throughout Section B we shall use homogeneous cardinals in defining the arithmetical operations: the arithmetical sum of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\), for example, will be defined by means of \(\text{N}_{0}\text{c}ʻ\alpha\) and \(\text{N}_{0}\text{c}ʻ\beta\), in order to exclude such a determination of the typical ambiguity of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) as would make either of them null. It is true that not only homogeneous cardinals, but also ascending cardinals (cf. *104), are never null. But homogeneous cardinals are much the simplest kind of cardinals that are never null, and are therefore the most convenient.

The fact that no homogeneous cardinal is null is derived from

*103·12. \(\vdash . \alpha \in \text{N}_{0}\text{c}ʻ\alpha\)

Other important propositions in this number are the following:

*103·2. \(\vdash : \mu \in \text{N}_{0}\text{C} . \equiv . (\exists \alpha ) . \mu = \text{Nc}ʻ\alpha \cap tʻ\alpha . \equiv . (\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha\)

*103·26. \(\vdash \colon\ldotp \mu \in \text{NC} . \supset : \alpha \in \mu . \equiv . \text{N}_{0}\text{c}ʻ\alpha = \mu\)

The above proposition is used constantly.

[Pg 37]

*103·27. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\equiv .\mu \in \text{NC}.\alpha \in \mu \)

Thus to say that \(\mu\) is the homogeneous cardinal of \(\alpha\) is equivalent to saying that \(\mu\) is a cardinal of which \(\alpha\) is a member.

*103·301. \(\vdash .\text{NC}^\alpha (\alpha )=\text{N}_{0}\text{C}(\alpha )\)

*103·34. \(\vdash .\text{NC} - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}\)

*103·4. \(\vdash .\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha\)

*103·41. \(\vdash .\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \cap tʻ\beta =\text{Nc}(\beta )ʻ\alpha \)

*103·01. \(\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻ\alpha \quad\text{Df}\)

*103·02. \(\text{N}_{0}\text{C}=\text{D}ʻ\text{N}_{0}\text{c} \quad\text{Df}\)

*103·1. \(\vdash .\text{N}_{0}\text{c}ʻ\alpha =(\text{Nc}ʻ\alpha )_{\alpha }=\text{Nc}(\alpha )ʻ\alpha =\text{Nc}(\alpha _{\alpha })ʻ\alpha \quad[*102·6.(*103·01)]\)

*103·11. \[\begin{aligned}&\vdash :\beta \in \text{N}_{0}\text{c}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \in tʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻ\alpha\\ &[*103·1.*102·6]\end{aligned}\]

*103·12. \(\vdash .\alpha \in \text{N}_{0}\text{c}ʻ\alpha \quad[*103·11.*73·3.*63·103]\)

*103·13. \(\vdash .\exists !\text{N}_{0}\text{c}ʻ\alpha \quad[*103·12.*10·24]\)

This is a legitimate inference from *103·12 because, when \(\alpha\) is given, \(\text{N}_{0}\text{c}ʻ\alpha\) is typically definite.

*103·14. \(\vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\equiv .\alpha \in \text{N}_{0}\text{c}ʻ\beta .\equiv .\beta \in \text{N}_{0}\text{c}ʻ\alpha .\equiv .\alpha \text{ sm }\beta .\alpha \in tʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*103·11.\supset \\ \vdash \colon\ldotp \text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\equiv :\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\equiv _{\gamma }.\gamma \text{ sm }\beta .\gamma \in tʻ\beta : &\qquad \text{(1)}\\ [*10·1] \supset :\alpha \text{ sm }\alpha .\alpha \in tʻ\alpha .\equiv .\alpha \text{ sm }\beta .\alpha \in tʻ\beta :\\ [*73·3.*63·103] \supset :\alpha \text{ sm }\beta .\alpha \in tʻ\beta &\qquad \text{(2)}\\ \vdash .*73·32.*63·17\\ \vdash :\alpha \text{ sm }\beta .\alpha \in tʻ\beta .\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\supset .\gamma \text{ sm }\beta .\gamma \in tʻ\beta &\qquad \text{(3)}\\ \vdash .(3)\frac{\beta,\,\alpha}{\alpha,\,\beta}.*73·31.*63·16.\supset \\ \vdash :\alpha \text{ sm }\beta .\alpha \in tʻ\beta .\gamma \text{ sm }\beta .\gamma \in tʻ\beta .\supset .\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha &\qquad \text{(4)}\\ \vdash .(3).(4).(1).\supset \\ \vdash :\alpha \text{ sm }\beta .\alpha \in tʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(5)}\\ \vdash .(2).(5).*103·11.*73·31.*63·16.\supset \vdash .\text{Prop} \end{array} \]

*103·15. \(\vdash :\exists !\text{N}_{0}\text{c}ʻ\alpha \cap \text{N}_{0}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*103·13.&\supset \vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\exists !\text{N}_{0}\text{c}ʻ\alpha \cap \text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ \vdash .*103·14.\supset \vdash :\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\gamma \in \text{N}_{0}\text{c}ʻ\beta .&\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\gamma .\\ [*14·131·144] &\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta :\\ [*10·11·23] &\supset \vdash :\exists !\text{N}_{0}\text{c}ʻ\alpha \cap \text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash .\text{Prop} \end{array} \]

[Pg 38]

*103·16. \(\vdash : \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta . \equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta \)

In this proposition, the equation "\(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)" must be supposed to hold in any type for which it is significant. Otherwise, we might find a type for which \(\text{Nc}ʻ\alpha = \Lambda = \text{Nc}ʻ\beta\), without having \(\text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta\).

Dem. \[ \begin{array}{l} \vdash . *103·12 . \supset \vdash : \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta . &\supset . \alpha \in \text{Nc}ʻ\beta .\\ [*100·31·321] &\supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta &\qquad \text{(1)}\\ \vdash . *22·481. \supset \vdash : \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . &\supset . \text{Nc}ʻ\alpha \cap tʻ\alpha = \text{Nc}ʻ\beta \cap tʻ\alpha .\\ [*65·13.(*103·01)] &\supset . \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2). \supset \vdash . \text{Prop} \end{array} \]

*103·2. \[\begin{aligned}&\vdash : \mu \in \text{N}_{0}\text{C} . \equiv . (\exists \alpha ) . \mu = \text{Nc}ʻ\alpha \cap tʻ\alpha . \equiv . (\exists \alpha ). \mu = \text{N}_{0}\text{c}ʻ\alpha\\ &[*71·41.*100·22.(*103·01·02)]\end{aligned}\]

*103·21. \(\vdash . \text{N}_{0}\text{c}ʻ\alpha \in \text{N}_{0}\text{C} . \text{N}_{0}\text{c}ʻ\alpha \in \text{NC} \quad[*103·2.*100·2·4.*14·28.*65·13]\)

In adducing a proposition, such as *100·2, which is concerned with an "\(\text{Nc}\)" entirely undetermined in type, any degree of typical determination may be added to our "\(\text{Nc}\)," since an asserted proposition containing an ambiguous "\(\text{Nc}\)" is only legitimate if it is true for every possible determination of the ambiguity.

*103·22. \(\vdash : \mu \in \text{N}_{0}\text{C} . \supset . \exists !\mu \quad[*103·13·2]\)

*103·23. \(\vdash . \Lambda {\sim}\in \text{N}_{0}\text{C} \quad[*103·22]\)

*103·24. \(\vdash . \text{N}_{0}\text{C} \in \text{Cls ex}^{2}\,\text{excl}\quad[*100·43.*103·23.*84·13]\)

*103·25. \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C} . \supset : \exists !\mu \cap \nu . \equiv . \mu = \nu \quad[*103·24 . *84·135]\)

*103·26. \(\vdash \colon\ldotp \mu \in \text{NC}. \supset : \alpha \in \mu . \equiv . \text{N}_{0}\text{c}ʻ\alpha = \mu\)

Dem. \[ \begin{array}{l} \vdash .*100·45. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \mu . \supset . \text{Nc}ʻ\alpha = \mu &\qquad \text{(1)}\\ \vdash .*63·22. &\supset \vdash : \alpha \in \mu . \supset . \mu \subset tʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*22·621. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \alpha \in \mu . \supset . \text{Nc}ʻ\alpha \cap tʻ\alpha = \mu .\\ [(*103·01)] &\supset . \text{N}_{0}\text{c}ʻ\alpha = \mu &\qquad \text{(3)}\\ \vdash . *103·12. &\supset \vdash : \text{N}_{0}\text{c}ʻ\alpha = \mu . \supset . \alpha \in \mu &\qquad \text{(4)}\\ \vdash .(3).(4). \supset \vdash . \text{Prop} \end{array} \]

*103·27. \(\vdash : \mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv . \mu \in \text{NC} . \alpha \in \mu\)

Dem. \[ \begin{array}{l} \vdash .*103·26. \supset \vdash : \mu \in \text{NC} . \mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv . \mu \in \text{NC} . \alpha \in \mu &\qquad \text{(1)}\\ \vdash .(1).*103·21. \supset \vdash . \text{Prop} \end{array} \]

*103·28. \(\vdash : (\exists \alpha ) . \gamma \text{ sm } \alpha . \mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv . \exists ! \mu . \mu = \text{Nc}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *103·27 . \supset \\ \vdash : (\exists \alpha ) . \gamma \text{ sm } \alpha . \mu = \text{N}_{0}\text{c}ʻ\alpha . &\equiv . (\exists \alpha ) . \gamma \text{ sm } \alpha . \mu \in \text{NC} . \alpha \in \mu .\\ [*100·31] &\equiv . \mu \in \text{NC} . \exists ! \mu \cap \text{Nc}ʻ\gamma .\\ [*100·42·41] &\equiv . \mu \in \text{NC} . \exists ! \mu \cap \text{Nc}ʻ\gamma . \mu = \text{Nc}ʻ\gamma .\\ [*100·41] &\equiv . \exists ! \mu . \mu = \text{Nc}ʻ\gamma : \supset \vdash . \text{Prop} \end{array} \]

[Pg 39]

*103·3. \(\vdash :\beta \in tʻ\alpha .\supset .\text{N}_{0}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\alpha })ʻ\beta =\text{Nc}ʻ\beta \cap tʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*63·16.\supset \vdash :\text{Hp}.\supset .tʻ\beta =tʻ\alpha .\\ [*22·481.(*103·01)] \supset .\text{N}_{0}\text{c}ʻ\beta &=\text{Nc}ʻ\beta \cap tʻ\alpha &\qquad \text{(1)}\\ [*102·6]&=\text{Nc}(\alpha )ʻ\beta &\qquad \text{(2)}\\ [*102·61] &=\text{Nc}(\alpha _{\alpha })ʻ\beta &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*103·301. \(\vdash .\text{NC}^\alpha (\alpha )=\text{N}_{0}\text{C}(\alpha )\)

Note that although "\(\text{NC}(\alpha )\)" is not definite, "\(\text{N}_{0}\text{C}(\alpha )\)" is absolutely definite as soon as \(\alpha\) is assigned.

Dem. \[ \begin{array}{l} \vdash .*103·3.\supset \vdash :\beta \in tʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\beta .\equiv .\beta \in tʻ\alpha .\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta .\\ [*102·37] \equiv .\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta &\qquad \text{(1)}\\ \vdash .*63·5.(*103·01).\supset \\ \vdash \colon\ldotp \mu =\text{N}_{0}\text{c}ʻ\beta .\supset :\beta \in tʻ\alpha .\equiv .\mu \in t^{2}ʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash :\mu \in t^{2}ʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\beta .\equiv .\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta &\qquad \text{(3)}\\ \vdash .(3).*10·11·281·35.\supset \\ \vdash \colon\ldotp \mu \in t^{2}ʻ\alpha :(\exists \beta ).\mu =\text{N}_{0}\text{c}ʻ\beta :\equiv .(\exists \beta ).\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta .\\ [*102·5] \equiv .\mu \in \text{NC}^\alpha (\alpha ) &\qquad \text{(4)}\\ \vdash .(4).*103·2.\supset \vdash :\mu \in t^{2}ʻ\alpha \cap \text{N}_{0}\text{C}.\equiv .\mu \in \text{NC}^\alpha (\alpha ) &\qquad \text{(5)}\\ \vdash .(5).(*65·02).\supset \vdash .\text{Prop} \end{array} \]

*103·31. \(\vdash :\exists !\text{Nc}(\alpha _{\beta })ʻ\delta .\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{N}_{0}\text{C}(\alpha )\)

Dem. \[ \begin{array}{l} \vdash .*102·52.\supset \vdash :\text{Hp}.&\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{NC}^\alpha (\alpha ).\\ [*103·301] &\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{N}_{0}\text{C}(\alpha ):\supset \vdash .\text{Prop} \end{array} \]

*103·32. \(\vdash .\text{NC}^\beta (\alpha )-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}(\alpha )\)

Dem. \[ \begin{array}{l} \vdash .*103·31.\supset \vdash :\mu =\text{Nc}(\alpha _{\beta })ʻ\delta .\exists !\mu .\supset .\mu \in \text{N}_{0}\text{C}(\alpha ) &\qquad \text{(1)}\\ \vdash .(1).*102·5.\supset \vdash .\text{Prop} \end{array} \]

In the above proposition, the "\(\beta\)" may be omitted, and we may write (cf. *103·33, below) \[ \vdash .\text{NC}(\alpha )-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}(\alpha ). \]

For the \(\beta\) is wholly arbitrary, so that any possible determination of \(\text{NC}(\alpha )\) makes the above proposition true. We may proceed a step further, and write (*103·34, below) \[ \vdash .\text{NC}-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}. \]

[Pg 40]

But although we also have \(\text{N}_{0}\text{C}\subset \text{NC}-\iotaʻ\Lambda\), provided the "\(\text{NC}\)" on the right is suitably determined, we do not have this always. For example, if "\(\text{NC}\)" is determined as \(\text{NC}^\alpha (tʻ\alpha )\), and "\(\text{N}_{0}\text{C}\)" as \(\text{N}_{0}\text{C}(tʻ\alpha )\), then \(\text{N}_{0}\text{c}ʻtʻ\alpha \in \text{N}_{0}\text{C}-\text{NC}\).

*103·33. \(\vdash . \text{NC}(\alpha ) - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}(\alpha )\)

Dem. \[ \begin{array}{l} \vdash . *4·2 . (*65·02) . \supset \\ \vdash \colon\ldotp \mu \in \text{NC}(\alpha ) - \iota ʻ\Lambda . \equiv : \mu \in \text{NC} . \mu \in t^{2}ʻ\alpha . \exists ! \mu :\\ [*100·4 . *63·5] \equiv : (\exists \beta ) . \mu = \text{Nc}ʻ\beta : \mu \subset tʻ\alpha . \exists ! \mu :\\ [*65·13] \equiv : (\exists \beta ) . \mu = \text{Nc}ʻ\beta \cap tʻ\alpha : \exists ! \mu :\\ [*102·6] \equiv : (\exists \beta ) . \mu = \text{Nc}(\alpha _{\beta })ʻ\beta . \exists ! \mu :\\ [*103·31] \supset : \mu \in \text{N}_{0}\text{C}(\alpha ) \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

*103·34. \(\vdash . \text{NC} - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}\)

Dem. \[ \begin{array}{l} \vdash . *100·31·321 . *63·5 . \supset \\ \vdash : \mu = \text{Nc}ʻ\alpha . \beta \in \mu . \supset . \mu = \text{Nc}ʻ\beta \cap tʻ\beta \\ [(*103·01)] = \text{N}_{0}\text{c}ʻ\beta .\\ [*103·2] \supset . \mu \in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ \vdash . (1) . *100·4 . *11·11·35·54 . \supset \vdash . \text{Prop} \end{array} \]

Thus every cardinal except \(\Lambda\) is a homogeneous cardinal in the appropriate type. Note that although of course every homogeneous cardinal is a cardinal, yet "\(\text{N}_{0}\text{C}\subset \text{NC}\)" must not be asserted, because it is possible to determine the ambiguity of "\(\text{NC}\)" in such a way as to make this false. Hence we do not get \(\text{NC} - \iota ʻ\Lambda =\text{N}_{0}\text{C}\).

*103·35. \(\vdash : \Lambda {\sim} \in \text{NC}^\alpha (\beta ) . \supset . \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda = \text{N}_{0}\text{C}(\alpha ) \quad[*102·55 . *103·301]\)

The hypothesis of this proposition is satisfied, as will appear later, if the type of \(\beta\) is in what we may call the direct ascent from that of \(\alpha\), i.e. if it can be reached from \(\alpha\) by a finite number of steps each of which takes us from a type \(\tau\) to either \(\text{Cl}ʻ\tau\) or \(\text{Rl}ʻ(\tau \uparrow \tau)\). Thus in such a case the cardinals (other than \(\Lambda\)) which go from \(tʻ\beta\) to \(tʻ\alpha\) are the same as those which begin and end within \(tʻ\alpha\). It will also appear that in such a case \(\Lambda\) always is a member of \(\text{NC}^\beta (\alpha )\). If two cardinals which are not equal must always be one greater and the other less, then \(\Lambda \in \text{NC}^\beta (\alpha )\) is the condition for \(\text{N}_{0}\text{c}ʻtʻ\beta \gt \text{Nc}(\beta )ʻtʻ\alpha\). In that case, we shall have \(\Lambda \in \text{NC}^\beta (\alpha ). \supset . \Lambda {\sim} \in \text{NC}^\alpha(\beta )\). But there is no known proof that of two different cardinals one must be the greater, except by assuming the multiplicative axiom and proving thence (by Zermelo's theorem) that every class can be well-ordered (cf. *258).

*103·4. \(\vdash . \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash . *37·1 . \supset \\ \vdash : \delta \in \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha . &\equiv . (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma .\\ [*102·84] &\equiv . \delta \text{ sm } \alpha : \supset \vdash . \text{Prop} \end{array} \]

*103·41. \(\vdash . \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \cap tʻ\beta = \text{Nc}(\beta )ʻ\alpha \)

Dem. \[ \begin{array}{l} \vdash . *103·4 . \supset \vdash .\\ \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \cap tʻ\beta &= \text{Nc}ʻ\alpha \cap tʻ\beta \\ [*102·6] &= \text{Nc}(\beta )ʻ\alpha . \supset \vdash . \text{Prop} \end{array} \]

[Pg 41]

*103·42. \(\vdash :\beta \text{ sm } \alpha . \equiv .\text{Nc}(\beta )ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta \)

Dem. \[ \begin{array}{l} \vdash .*100·321.\supset \vdash :\beta \text{ sm } \alpha .&\supset .\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta .\\ [*22·481] &\supset .\text{Nc}ʻ\alpha \cap tʻ\beta = \text{Nc}ʻ\beta \cap tʻ\beta .\\ [*102·6.(*103·01)] &\supset .\text{Nc}(\beta )ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ \vdash .*103·12.\supset \vdash :\text{Nc}(\beta )ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta .&\supset .\beta \in \text{Nc}(\beta )ʻ\alpha .\\ [*100·31] &\supset .\beta \text{ sm } \alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*103·43. \(\vdash :\mu \in \text{NC}.\supset .\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu = \mu\)

Dem. \[ \begin{array}{l} \vdash .*37·29. &\supset \vdash :\mu = \Lambda .\supset .\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu = \Lambda &\qquad \text{(1)}\\ \vdash .*103·27. \supset \vdash :\mu \in \text{NC}.\alpha \in \mu .&\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha .t_{0}ʻ\mu = tʻ\alpha .\\ [*103·41] &\supset .\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu = \text{Nc}(\alpha )ʻ\alpha \\ [*103·3·27] &= \mu &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*103·44. \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\supset :\mu = \text{ sm }ʻʻ\nu .\equiv .\nu = \text{ sm }ʻʻ\mu\)

Dem. \[ \begin{array}{l} \vdash .*100·53. &\supset \vdash \colon\ldotp \exists !\mu .\exists !\nu .\mu ,\nu \in \text{NC}.\supset :\mu = \text{ sm }ʻʻ\nu .\equiv .\nu = \text{ sm }ʻʻ\mu &\qquad \text{(1)}\\ \vdash .*103·27·2.&\supset \vdash :\text{Hp}.\supset .\exists !\mu .\exists !\nu .\mu ,\nu \in \text{NC} &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash .\text{Prop} \end{array} \]

*103.5. \(\vdash .0\in \text{N}_{0}\text{C}\)

Dem. \[ \begin{array}{l} \vdash .*101·11·12.&\supset \vdash .0\in \text{NC}.\exists !0.\\ [*103·34]&\supset \vdash .0\in \text{N}_{0}\text{C}.\supset \vdash .\text{Prop} \end{array} \]

*103·51. \(\vdash .1\in \text{N}_{0}\text{C}\)

Dem. \[ \begin{array}{l} \vdash .*101·21·241.&\supset \vdash .1\in \text{NC}.\exists !1.\\ [*103·34] &\supset \vdash .1\in \text{N}_{0}\text{C}.\supset \vdash .\text{Prop} \end{array} \]

0 and 1 are the only cardinals of which the above property can be proved universally with our assumptions. If (as is possible so far as our assumptions go) the lowest type is a unit class, we shall have in that type (though in no other) \(2 = \Lambda\), so that in that type \(2 {\sim} \in \text{N}_{0}\text{C}\).

Summary of *104.

In this number we have to consider cardinals derived from a relation of similarity which goes from the type of \(\alpha\) to that of \(tʻ\alpha\), or to that of \(t^{2}ʻ\alpha\). The propositions to be proved can be extended, by a mere repetition of the proofs, to \(t^{3}ʻ\alpha\), \(t^{4}ʻ\alpha\), etc. This extension must, however, be made afresh in each instance; we cannot prove that it can be made generally, because mathematical induction cannot be applied to the series \[ t_{0}ʻ\alpha,\quad tʻ\alpha,\quad t^{2}ʻ\alpha,\quad t^{3}ʻ\alpha ,\ldotp\ldots \]

Ascending cardinals, though less important than homogeneous cardinals, yet have considerable importance in arithmetic, because \(\text{Nc}ʻ\alpha \times \text{Nc}ʻ\beta\) and \((\text{Nc}ʻ\alpha)^{\text{Nc}ʻ\beta }\) are defined as the cardinals of classes of higher types than those of \(\alpha\) and \(\beta\), and the same applies to the product of the cardinals of members of a class of classes. In these cases, however, we also need cardinals of relational types, which will be dealt with in *106.

We have to deal, in this number, with three different sets of notions, namely

*104·01. \(\text{N}^{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻtʻ\alpha \quad\text{Df}\)

*104·02. \(\text{N}^{1}\text{C}=\text{D}ʻ\text{N}^{1}\text{c} \quad\text{Df}\)

*104·03. \(\mu ^{(1)}=\text{ sm }ʻʻ\mu \cap tʻ\mu \quad\text{Df}\)

with similar definitions of \(\text{N}^{2}\text{c}ʻ\alpha\), etc. Thus \(\text{N}^{1}\text{c}ʻ\alpha\) consists of all classes similar to \(\alpha\) but of the next higher type, i.e. it is the cardinal number of \(\alpha\) in the type next above that of \(\text{N}_{0}\text{c}ʻ\alpha\); \(\text{N}^{1}\text{C}\) is the class of all such cardinals as \(\text{N}^{1}\text{c}ʻ\alpha\), and is a typically ambiguous symbol, though \(\text{N}^{1}\text{c}ʻ\alpha\) is typically definite when \(\alpha\) is given; \(\mu ^{(1)}\) (if \(\mu\) is a cardinal which is not null) is the "same" cardinal in the next higher type, so that, e.g., if \(\mu\) is 1 determined as consisting of unit classes of individuals, \(\mu ^{(1)}\) will be 1 determined as consisting of unit classes of classes of individuals. (When \(\mu\) is not an existent cardinal, \(\mu ^{(1)}\) is unimportant.)

The following are the most useful propositions in the present number:

*104·12. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\gamma \in \text{N}^{1}\text{c}ʻ\beta .\supset .\gamma \in \text{N}^{2}\text{c}ʻ\alpha\)

[Pg 43]

*104·2. \(\vdash .ʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha\)

*104·21. \(\vdash .\exists !\text{N}^{1}\text{c}ʻ\alpha\)

*104·24. \(\vdash :\mu = \text{N}^{1}\text{c}ʻ\alpha .\supset .\mu = \text{N}_{0}\text{c}ʻ\iota ʻʻ\alpha = \text{N}_{0}\text{c}ʻ\hat{\beta} \{(\exists y).y\in \alpha .\beta = \iota ʻx\cup \iota ʻy\}\)

*104·25. \(\vdash .\text{N}^{1}\text{C}\subset \text{C}\text{N}_{0}\text{C}\)

*104·26. \(\vdash :\mu = \text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(1)} = \text{N}_{0}\text{c}ʻ\iota ʻʻ\alpha = \text{N}^{1}\text{c}ʻ\alpha\)

*104·265. \(\vdash .\mu ^{(1)} = \text{ sm }_{\mu }ʻʻ\mu\)

*104·27. \(\vdash \colon\ldotp \mu \in \text{NC}.\supset :\mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv .\mu ^{(1)} = \text{N}^{1}\text{c}ʻ\alpha\)

*104·35. \(\vdash .\text{N}^{2}\text{C}\subset \text{N}^{1}\text{C}.\text{N}^{2}\text{C}\subset \text{N}_{0}\text{C}\)

*104·43. \(\vdash :tʻ\alpha = tʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda\)

*104·01. \(\text{N}^{1}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻtʻ\alpha \quad\text{Df}\)

This defines the cardinal number of \(\alpha\) in the next type above that of \(\text{N}_{0}\text{c}ʻ\alpha\); thus \(\text{N}^{1}\text{c}ʻ\alpha\) consists of all classes similar to \(\alpha\) and of the next type above that of \(\alpha\).

*104·011. \(\text{N}^{2}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt^{2}ʻ\alpha \quad\text{Df}\)

Similar definitions are to be assumed for \(\text{N}^{3}\text{c}ʻ\alpha\), etc.

*104·02. \(\text{N}^{1}\text{C} = \text{D}ʻ\text{N}^{1}\text{c} \quad\text{Df}\)

\(\text{N}^{1}\text{C}\), like \(\text{N}_{0}\text{C}\), is typically ambiguous; but \(\text{N}^{1}\text{C}(\alpha )\) is typically definite.

*104·021. \(\text{N}^{2}\text{C} = \text{D}ʻ\text{N}^{2}\text{c} \quad\text{Df}\)

Similar definitions are to be assumed for \(\text{N}^{3}\text{C}\), etc.

*104·03. \(\mu ^{(1)} = \text{ sm }ʻʻ\mu \cap tʻ\mu \quad\text{Df}\)

Here, if \(\mu\) is a cardinal, \(\mu ^{(1)}\) is the same cardinal in the next higher type. For example, if \(\mu\) is couples of individuals, \(\mu ^{(1)}\) is couples of classes of individuals.

*104·031. \(\mu ^{(2)} = \text{ sm }ʻʻ\mu \cap t^{2}ʻ\mu \quad\text{Df}\)

Similar definitions are to be assumed for \(\mu ^{(3)}\), etc.

*104·1. \[\begin{aligned}&\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha . \equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻtʻ\alpha . \equiv .\beta \in \text{Nc}ʻ\alpha .\beta \subset tʻ\alpha \\ &[*63·5. (*104-01)]\end{aligned}\]

*104·101. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha . \equiv .\beta \text{ sm } \alpha .\beta \subset tʻ\alpha \quad[*100·31.*104·1]\)

*104·102. \(\vdash .\text{N}^{1}\text{c}ʻ\alpha = \text{Nc}(tʻ\alpha )ʻ\alpha = \text{Nc} {(tʻ\alpha )_{\alpha }}ʻ\alpha \quad[*102·6.(*104·01)]\)

*104·11. \[\begin{aligned}&\vdash :\beta \in \text{N}^{2}\text{c}ʻ\alpha . \equiv .\beta \in \text{Nc}ʻ\alpha . \beta \in tʻt^{2}ʻ\alpha . \equiv .\beta \in \text{Nc}ʻ\alpha .\beta \subset t^{2}ʻ\alpha\\ &[*63·5.(*104·011)]\end{aligned}\]

*104·111. \(\vdash :\beta \in \text{N}^{2}\text{c}ʻ\alpha . \equiv .\beta \text{ sm } \alpha .\beta \subset t^{2}ʻ\alpha \quad[*100·31.*104·11]\)

[Pg 44]

*104·112. \(\vdash .\text{N}^{2}\text{c}ʻ\alpha = \text{Nc}(t^{2}ʻ\alpha )ʻ\alpha = \text{Nc}{(t^{2}ʻ\alpha )_{\alpha }}ʻ\alpha \quad[*102·6.(*104·011)]\)

*104·12. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\gamma \in \text{N}^{1}\text{c}ʻ\beta .\supset .\gamma \in \text{N}^{2}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·1.\supset \vdash :\text{Hp}.&\supset .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻtʻ\alpha .\gamma \in \text{Nc}ʻ\beta .\gamma \in tʻtʻ\beta .\\ [*100·32] &\supset .\gamma \in \text{Nc}ʻ\alpha .\beta \in tʻtʻ\alpha .\gamma \in tʻtʻ\beta .\\ [*63·16] &\supset .\gamma \in \text{Nc}ʻ\alpha .tʻ\beta = tʻtʻ\alpha .\gamma \in tʻtʻ\beta .\\ [*13·12] &\supset .\gamma \in \text{Nc}ʻ\alpha .\gamma \in tʻtʻtʻ\alpha .\\ [*104·11] &\supset .\gamma \in \text{N}^{2}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*104·121. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\gamma \in \text{N}^{2}\text{c}ʻ\alpha .\supset .\gamma \in \text{N}^{1}\text{c}ʻ\beta \)

Dem. \[ \begin{array}{l} \vdash .*104·102·112.\supset \vdash :\text{Hp}.&\supset .\beta \in \text{Nc}\{(tʻ\alpha)_{\alpha}\} ʻ\alpha .\gamma \in \text{Nc}\{(t^{2}ʻ\alpha)_{\alpha}\}ʻ\alpha.\\ [*102·41] &\supset .\gamma \in \text{Nc}\{(t^{2}ʻ\alpha )_{tʻ\alpha}\}ʻ\beta &\qquad \text{(1)}\\ \vdash .*104·1.\supset \vdash :\text{Hp}.&\supset .\beta \in tʻtʻ\alpha .\\ [*63*16] &\supset .tʻ\beta = tʻtʻ\alpha .\\ [(*65·11)] &\supset .\text{Nc}\{(t^{2}ʻ\alpha )_{tʻ\alpha}\} = \text{Nc}\{(tʻ\beta)_{\beta}\} &\qquad \text{(2)}\\ \vdash .(1).(2).*104·102.\supset \vdash .\text{Prop} \end{array} \]

*104·122. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\text{N}^{1}\text{c}ʻ\beta = \text{N}^{2}\text{c}ʻ\alpha \quad[*104·12·121]\)

*104·123. \(\vdash :\text{N}_{0}\text{c}ʻ\beta = \text{N}^{1}\text{c}ʻ\alpha .\supset . \text{N}^{1}\text{c}ʻ\beta = \text{N}^{2}\text{c}ʻ\alpha \quad[*104·122.*103·26]\)

*104·13. \(\vdash :\mu \in \text{N}^{1}\text{C}. \equiv .(\exists \alpha ).\mu = \text{N}^{1}\text{c}ʻ\alpha \quad[*100·122.*71·41.(*104·02)]\)

*104·14. \[\begin{aligned}&\vdash :\delta \in \mu ^{(1)}. \equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm } \gamma .\delta \in tʻ\mu . \equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm } \gamma .\delta \subset tʻ\gamma\\ &[*37·1.*63·22.(*104·03)]\end{aligned}\]

*104·141. \(\vdash :\mu \in \text{NC}.\exists !\mu .\supset .\mu ^{(1)}\in \text{NC} \quad[*100·52]\)

When the hypothesis "\(\exists !\mu\)" is omitted, this proposition is still true, but with a difference. E.g. let us put \[ \mu = \text{Nc}(\alpha )ʻtʻ\alpha . \] Then \(\mu = \Lambda .\mu ^{(1)} = \Lambda\). Thus \(\mu^{(1)} \neq \text{Nc}(tʻ\alpha)ʻtʻ\alpha\). But we still have \[ \mu ^{(1)} = \text{Nc}(tʻ\alpha )ʻt^{2}ʻ\alpha . \] Thus \(\mu^{(1)}\in \text{NC}\), but \(\mu^{(1)}\) is not the same cardinal as \(\mu\) in a higher type, i.e. there are classes whose cardinal in one type is \(\mu\), but whose cardinal in the next higher type is not \(\mu ^{(1)}\).

*104·142. \(\vdash :\mu \in \text{NC}.\exists !\mu .\supset .\mu ^{(2)}\in \text{NC} \quad[*100·52]\)

*104·15. \(\vdash :\mu \in \text{N}^{2}\text{C}. \equiv .(\exists \alpha ).\mu = \text{N}^{2}\text{c}ʻ\alpha \quad[*100·22.*71·41. (*104·021)]\)

*104·2. \(\vdash .\iota ʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*63·621.&\supset \vdash x\in \alpha .\supset _{x}.\iota ʻx\in tʻ\alpha :\\ [*37·61] &\supset \vdash .\iotaʻʻ\alpha \subset tʻ\alpha &\qquad \text{(1)}\\ \vdash .(1).*100·6.*104·1.\supset \vdash . \text{Prop} \end{array} \]

[Pg 45]

*104·201. \(\vdash :\beta \in \text{N}_{0}\text{c}ʻ\alpha .\supset .\iotaʻʻ\beta \in \text{N}^{1}\text{c}ʻ\alpha .\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*100·31·321.\supset \vdash :\text{Hp}.&\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta &\qquad \text{(1)}\\ \vdash .*103·11. \supset \vdash :\text{Hp}.&\supset .\beta \in tʻ\alpha .\\ [*63·16] &\supset .tʻ\alpha =tʻ\beta .\\ [*30·37] &\supset .tʻtʻ\alpha =tʻtʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).(*104·01).&\supset \vdash .\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta &\qquad \text{(3)}\\ \vdash .(3).*104·2.\supset \vdash .\text{Prop} \end{array} \]

*104·21. \(\vdash .\exists !\text{N}^{1}\text{c}ʻ\alpha \quad[*104·2]\)

It follows from this proposition that ascending cardinals are never null. The proof has to be made separately for each kind of ascending cardinal, i.e. \(\text{N}^{1}\text{C}\), \(\text{N}^{2}\text{C}\), etc.

*104·211. \(\vdash .\exists !\text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ1 \quad[*104·2.*52·3]\)

*104·23. \(\vdash .\hat{\beta} \{(\exists y).y\in \alpha .\beta =\iota ʻx\cup \iota ʻy\}\in \text{N}^{1}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*51·16. &\supset \vdash :y\in \alpha .\supset y\in \alpha \cap (\iota ʻx\cup \iota ʻy).\\ [*63·16] &\supset .\iota ʻx\cup \iota ʻy\in tʻ\alpha &\qquad \text{(1)}\\ \vdash .(1).*10·11·23.&\supset \vdash .\hat{\beta} \{(\exists y).y\in \alpha .\beta =\iota ʻx\cup \iota ʻy\}\subset tʻ\alpha &\qquad \text{(2)}\\ \vdash .(2).*100·61.*104·1.\supset \vdash .\text{Prop} \end{array} \]

*104·231. \(\vdash :\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*104·2.\supset \vdash :\text{Hp}.&\supset .\iotaʻʻ\beta \in \text{N}_{1}\text{c}ʻ\alpha .\\ [*104·101] &\supset .\iotaʻʻ\beta \text{ sm }\alpha .\iotaʻʻ\beta \subset tʻ\alpha .\\ [*73·41.*63·21·64] &\supset .\beta \text{ sm }\alpha .tʻ\beta =tʻ\alpha .\\ [*103·11.*63·16] &\supset .\beta \in \text{N}_{0}\text{c}ʻ\alpha .\\ [*103·14] &\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta :\supset \vdash .\text{Prop} \end{array} \]

*104·232. \[\begin{aligned}&\vdash :\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\equiv .\beta \in \text{N}_{0}\text{c}ʻ\alpha\\ &[*104·231·201.*103·14]\end{aligned}\]

*104·24. \[\begin{aligned}&\vdash :\mu =\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\iotaʻʻ\alpha =\text{N}_{0}\text{c}ʻ\hat{\beta} \{(\exists y).y\in \alpha .\beta =\iota ʻx\cup \iota ʻy\}\\ &[*104·2·23.*103·26]\end{aligned}\]

*104·25. \(\vdash .\text{N}^{1}\text{C}\subset \text{N}_{0}\text{C} \quad[*104·24·13]\)

This proposition holds for each possible determination of the typical ambiguities, i.e. for every \(\alpha\) we have \[ \begin{aligned} \text{N}^{1}\text{C}(tʻ\alpha )&\subset \text{N}_{0}\text{C}(tʻ\alpha ).\\ \text{We do not have}\qquad \text{N}^{1}\text{C}(tʻ\alpha )&=\text{N}_{0}\text{C}(tʻ\alpha ),\\ \text{because}\qquad\qquad\qquad \text{N}_{0}\text{c}ʻtʻ\alpha &\in \text{N}_{0}\text{C}(tʻ\alpha )-\text{N}^{1}\text{C}(tʻ\alpha ). \end{aligned} \]

[Pg 46]

*104·251. \(\vdash .\Lambda {\sim}\in \text{N}^{1}\text{C} \quad[*104·25.*103·23]\)

*104·252. \(\vdash .\text{N}^{1}\text{C}\in \text{Cls ex}^{2}\,\text{excl} \quad[*104·25.*103·24.*84·26]\)

*104·26. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(1)}=\text{N}_{0}\text{c}ʻ\iotaʻʻ\alpha =\text{N}^{1}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·14.*103·11.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\delta \in \mu ^{(1)}.&\equiv .(\exists \gamma ).\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\delta \text{ sm }\gamma .\delta \subset tʻ\gamma . &\qquad \text{(1)}\\ [*73·32.*63·16] &\supset .\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ [*104·101] &\supset .\delta \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(2)}\\ \vdash .*104·101.\supset \\ \vdash :\delta \in \text{N}^{1}\text{c}ʻ\alpha . &\supset .\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ [*73·3.*63·103]&\supset .\alpha \text{ sm }\alpha .\alpha \in tʻ\alpha .\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ [*10·24] &\supset .(\exists \gamma ).\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\delta \text{ sm }\gamma .\delta \subset tʻ\gamma &\qquad \text{(3)}\\ \vdash .(3).(1).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\delta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\delta \in \mu ^{(1)} &\qquad \text{(4)}\\ \vdash .(2).(4).*104·24.\supset \vdash .\text{Prop} \end{array} \]

*104·261. \(\vdash :\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu \subset \text{N}_{0}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·14·101.\supset \\ \vdash \colon\ldotp \text{Hp}. \supset :(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta &\subset tʻ\gamma .\equiv _{\delta }.\delta \text{ sm }\alpha .\delta \subset tʻ\alpha :\\ [*10·23] \supset :\gamma \in \mu .\delta \text{ sm }\gamma .\delta \subset tʻ\gamma .&\supset _{\gamma ,\delta }.\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ [*4·7] &\supset _{\gamma ,\delta }.\delta \text{ sm }\alpha .\delta \text{ sm }\gamma .\delta \subset tʻ\alpha .\delta \subset tʻ\gamma .\\ [*73·32.*63·13] &\supset _{\gamma ,\delta }.\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\\ [*103·11] &\supset _{\gamma ,\delta }.\gamma \in \text{N}_{0}\text{c}ʻ\alpha &\qquad \text{(1)}\\ \vdash .(1).*10·23·35.*104·101.\supset \\ \vdash \colon\ldotp \text{Hp}. \supset :\gamma \in \mu .\exists !\text{N}^{1}\text{c}ʻ\gamma .\supset _{\gamma }.\gamma \in \text{N}_{0}\text{c}ʻ\alpha :\\ [*104·21] \supset :\gamma \in \mu .\supset _{\gamma }.\gamma \in \text{N}_{0}\text{c}ʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*104·262. \(\vdash :\mu \in \text{NC}.\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·21.\supset \vdash :\text{Hp}.&\supset .\exists !\mu ^{(1)}.\\ [*37·29.\text{Transp}] &\supset .\exists !\mu &\qquad \text{(1)}\\ \vdash .*103·26.&\supset \vdash :\text{Hp}.\gamma \in \mu .\supset .\mu =\text{N}_{0}\text{c}ʻ\gamma &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash :\text{Hp}.&\supset .(\exists \gamma ).\mu =\text{N}_{0}\text{c}ʻ\gamma .\\ [*104·26.\text{Hp}] &\supset .(\exists \gamma ).\mu =\text{N}_{0}\text{c}ʻ\gamma .\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma .\\ [*104·231] &\supset .(\exists \gamma ).\mu =\text{N}_{0}\text{c}ʻ\gamma .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\\ [*13·172] &\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*104·263. \(\vdash :\alpha \in \mu .\supset .\iotaʻʻ\alpha \in \mu ^{(1)}\)

Dem. \[ \begin{array}{l} \vdash .*73·41.*37·1.&\supset \vdash :\text{Hp}.\supset .\iotaʻʻ\alpha \in \text{ sm }ʻʻ\mu &\qquad \text{(1)}\\ \vdash .*63·64. &\supset \vdash :\text{Hp}.\supset .\iotaʻʻ\alpha \in tʻ\mu &\qquad \text{(2)}\\ \vdash .(1).(2).(*104·03).\supset \vdash .\text{Prop} \end{array} \]

[Pg 47]

*104·264. \(\vdash :\exists !\mu .\equiv .\exists !\mu ^{(1)}\)

Dem. \[ \begin{array}{l} \vdash .*104·263. &\supset \vdash :\exists !\mu .\supset .\exists !\mu ^{(1)} &\qquad \text{(1)}\\ \vdash .*37·29.\text{Transp}.(*104·03).&\supset \vdash :\exists !\mu ^{(1)}.\supset .\exists !\mu &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*104·265. \(\vdash .\mu ^(1)=\text{ sm }_\muʻʻ\mu \quad[*102·85.(*104·03)]\)

*104·27. \(\vdash \colon\ldotp \mu \in \text{NC}.\supset :\mu =\text{N}_{0}\text{c}ʻ\alpha .\equiv .\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha \quad[*104·26·262]\)

*104·28. \(\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\supset .\mu ^{(1)}\in \text{N}^{1}\text{C} \quad[*104·26.*103·34]\)

*104·29. \(\vdash :\nu \in \text{N}^{1}\text{C}.\equiv .(\exists \mu )·\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)}\)

Dem. \[ \begin{array}{l} \vdash .*104·26. \supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\mu ^{(1)}.\supset .\nu =\text{N}^{1}\text{c}ʻ\alpha :\\ [*10·11·28] \supset \vdash :(\exists \alpha ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\mu ^{(1)}.\supset .(\exists \alpha ).\nu =\text{N}^{1}\text{c}ʻ\alpha :\\ [*103·2.*104·13] \supset \vdash :\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)}.\supset .\nu \in \text{N}^{1}\text{C} &\qquad \text{(1)}\\ \vdash .*104·26.*103·2.\supset \\ \vdash :\nu =\text{N}^{1}\text{c}ʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\nu =\mu ^{(1)}.\mu \in \text{N}_{0}\text{C} &\qquad \text{(2)}\\ \vdash .(2).*10·11·28·35.\supset \\ \vdash \colon\ldotp \nu =\text{N}^{1}\text{c}ʻ\alpha :(\exists \mu ).\mu =\text{N}_{0}\text{c}ʻ\alpha :\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)} &\qquad \text{(3)}\\ \vdash .(3).*100·2.*14·204.\supset \\ \vdash :\nu =\text{N}^{1}\text{c}ʻ\alpha .\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)} &\qquad \text{(4)}\\ \vdash .(4).*10·11·23.*104·13.\supset \\ \vdash :\nu \in \text{N}^{1}\text{C}.\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)} &\qquad \text{(5)}\\ \vdash .(1).(5).\supset \vdash .\text{Prop} \end{array} \]

*104·3. \(\vdash .\iotaʻʻ\iotaʻʻ\alpha \in \text{N}^{2}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·2.&\supset \vdash .\iotaʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha .\iotaʻʻ\iotaʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha .\\ [*104·12] &\supset \vdash .\iotaʻʻ\iotaʻʻ\alpha \in \text{N}^{2}\text{c}ʻ\alpha \end{array} \]

*104·31. \(\vdash .\exists !\text{N}^{2}\text{c}ʻ\alpha \quad[*104·3]\)

*104·311. \(\vdash .\text{N}^{2}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha =\text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha \quad[*104·3·2.*103·26]\)

*104·32. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(2)}=\text{N}_{0}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha =\text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha =\text{N}^{2}\text{c}ʻ\alpha =\{\mu ^{(1)}\}^{(1)}\)

Dem. \[ \begin{array}{l} \vdash .*104·26.\supset \vdash :\text{Hp}.\supset .\{\mu ^{(1)}\}^{(1)}&=\text{N}_{0}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha &\qquad \text{(1)}\\ [*104·311] &=\text{N}^{2}\text{c}ʻ\alpha &\qquad \text{(2)}\\ \vdash .*103·11.(*104·031).\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\delta \in \mu ^{(2)}.&\equiv .(\exists \gamma ).\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\delta \text{ sm }\gamma .\delta \in tʻt^{2}ʻ\gamma .\\ [*102·84.*63·16] &\equiv .\delta \text{ sm }\alpha .\delta \in tʻt^{2}ʻ\alpha .\\ [*104·11] &\equiv .\delta \in \text{N}^{2}\text{c}ʻ\alpha &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*104·24.\supset \vdash .\text{Prop} \end{array} \]

[Pg 48]

*104·33. \(\vdash \colon\ldotp \mu \in \text{NC}.\supset :\mu =\text{N}_{0}\text{c}ʻ\alpha .\equiv .\mu ^{(2)}=\text{N}^{2}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·27.\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu =\text{N}_{0}\text{c}ʻ\alpha .&\equiv .\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha .\\ [*104·24] &\equiv .\mu ^{(1)}=\text{N}_{0}\text{c}ʻ\iotaʻʻ\alpha .\\ [*104·27·141.*103·13] &\equiv .\{\mu ^{(1)}\}^{(1)}=\text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha .\\ [*104·32·24] &\equiv .\mu ^{(2)}=\text{N}^{2}\text{c}ʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*104·34. \(\vdash :\varpi \in \text{N}^{2}\text{C}.\equiv .(\exists \nu ).\nu \in \text{N}^{1}\text{C}.\varpi =\nu ^{(1)}.\equiv .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)}\)

Dem. \[ \begin{array}{l} \vdash .*104·32.\supset \\ \vdash :\varpi =\text{N}^{2}\text{c}ʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\varpi =\mu ^{(2)}.\mu \in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ \vdash .(1).*100·2.*10·11·28·35.\supset\\ \vdash :(\exists \alpha ).\varpi =\text{N}^{2}\text{c}ʻ\alpha .\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)} &\qquad \text{(2)}\\ \vdash *104·32. \supset \vdash :\mu =\text{N}_{0}\text{C}ʻ\alpha .\varpi =\mu ^{(2)}.\supset .\varpi =\text{N}^{2}\text{c}ʻ\alpha .\\ [*104·15.*103·2]\supset \vdash :\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)}.\supset .\varpi \in \text{N}^{2}\text{C} &\qquad \text{(3)}\\ \vdash (2).(3). \supset \vdash :\varpi \in \text{N}^{2}\text{C}.\equiv .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)}. &\qquad \text{(4)}\\ [*104·32] \equiv .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\{\mu ^{(1)}\}^{(1)}.\\ [*13·195] \equiv .(\exists \mu ,\nu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)}.\varpi =\nu^{(1)}. [*104·29] \equiv .(\exists \nu).\nu \in \text{N}^{1}\text{C}.\varpi =\nu ^{(1)} &\qquad \text{(5)}\\ \vdash .(4).(5).\supset \vdash .\text{Prop} \end{array} \]

*104·35. \(\vdash .\text{N}^{2}\text{C}\subset \text{N}^{1}\text{C}.\text{N}^{2}\text{C}\subset \text{N}_{0}\text{C} \quad[*104·311·13·15]\)

*104·36. \(\vdash :\gamma \in \text{N}^{2}\text{c}ʻ\alpha .\gamma \in \text{N}^{1}\text{c}ʻ\beta .\supset .\beta \in \text{N}^{1}\text{c}ʻ\alpha .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*104·1·11.\supset \vdash :\text{Hp}.&\supset .\gamma \in \text{Nc}ʻ\alpha .\gamma \in tʻt^{2}ʻ\alpha .\gamma \in \text{Nc}ʻ\beta .\gamma \in tʻtʻ\beta .\\ [*100·34.*63·16] & \supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .tʻt^{2}ʻ\alpha =tʻtʻ\beta .\\ [*63·35·15] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .t^{2}ʻ\alpha =tʻ\beta .\\ [(*104·01.*103·01)] &\supset .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*103·12.\supset \vdash .\text{Prop} \end{array} \]

*104·37. \(\vdash :\text{N}^{2}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .\equiv .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*104·21.\supset \vdash :\text{N}^{2}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .&\supset .\exists !\text{N}^{2}\text{c}ʻ\alpha \cap \text{N}^{1}\text{c}ʻ\beta.\\ [*104·36] &\supset .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*104·123.\supset \vdash .\text{Prop} \end{array} \]

The following propositions are concerned with the proof that, given any two cardinals \(\mu\) and \(\nu\), of the same type, we can find two mutually exclusive classes one of which has \(\mu\) terms while the other has \(\nu\) terms. The proof requires that we should raise the types of both \(\mu\) and \(\nu\) one degree above[Pg 49] that in which they were originally given, i.e. that we should turn \(\mu\) and \(\nu\) into \(\mu ^{(1)}\) and \(\nu ^{(1)}\). Thus, for example, suppose the total number of individuals in the universe were finite (a supposition which is consistent with our primitive propositions), and suppose \(\mu\) were this number. Then unless \(\nu =0\), a class of \(\nu\) individuals will be an existent sub-class of the only class which consists of \(\mu\) individuals, and therefore we shall have \[ \alpha \in \mu .\beta \in \nu .\supset _{\alpha ,\beta }.\exists !\alpha \cap \beta . \]

But if we consider classes of \(\mu\) classes and \(\nu\) classes, we shall always be able to find a \(\gamma\) and a \(\delta\) such that \[ \gamma \in \mu ^{(1)}.\delta \in \nu^{(1)}.\gamma \cap \delta =\Lambda . \]

The existence of such a \(\gamma\) and \(\delta\) is important in connection with the arithmetical operations, and is therefore proved here.

*104·4. \[\begin{aligned}\vdash \colon\ldotp x\in \alpha .x\neq y.x\neq z.y\neq z:&(w).w_{\iota }=\hat{\alpha}\hat{u}(\alpha =\iota ʻw\cup \iota ʻu):\supset .\\ &x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*100·61. &\supset \vdash :\text{Hp}.\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\text{ sm }(\alpha -\iota ʻx) &\qquad \text{(1)}\\ \vdash .*73·43. &\supset \vdash :\text{Hp}.\supset .\iota ʻy_{\iota }ʻz\text{ sm }\iota ʻx &\qquad \text{(2)}\\ \vdash .*51·232.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .x{\sim}\in y_{\iota }ʻz &\qquad \text{(3)}\\ \vdash .*51·232. &\supset \vdash :\text{Hp}.\gamma \in x_{\iota }ʻʻ(\alpha -\iota ʻx).\supset .x\in \gamma &\qquad \text{(4)}\\ \vdash .(3).(4). \supset \vdash :\text{Hp}.&\supset .y_{\iota }ʻz{\sim}\in x_{\iota }ʻʻ(\alpha -\iota ʻx).\\ [*51·211] &\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cap \iota ʻy_{\iota }ʻz=\Lambda &\qquad \text{(5)}\\ \vdash .*51·21·211. &\supset \vdash .(\alpha -\iota ʻx)\cap \iota ʻx=\Lambda &\qquad \text{(6)}\\ \vdash .(1).(2).(5).(6).*73·71.*51·221.\supset \\ \vdash :\text{Hp}.&\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\text{ sm }\alpha &\qquad \text{(7)}\\ \vdash .*63·101·16.*51·232·16.\supset \\ \vdash :\text{Hp}. &\supset .tʻx=tʻy.x\in \alpha .y\in y_{\iota }ʻz.y_{\iota }ʻz\in x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz.\\ [*63·53·2] &\supset .t^{2}ʻx=tʻ\alpha .t^{2}ʻy=t_{0}ʻ\{x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota}ʻz\}.t^{2}ʻx=t^{2}ʻy.\\ [*13·17] &\supset .tʻ\alpha =t_{0}ʻ\{x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\}.\\ [*63·105] &\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\subset tʻ\alpha &\qquad \text{(8)}\\ \vdash .*54·26.&\supset \vdash :\text{Hp}.\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\subset 2 &\qquad \text{(9)}\\ \vdash .(7).(8).(9).*104·101.\supset \vdash .\text{Prop} \end{array} \]

*104·41 \[\begin{aligned}\vdash \colon\ldotp tʻ\alpha =tʻ\beta :(\exists x,y,z).x\in \alpha .&x\neq y.x\neq z.y\neq z:\supset .\\ &(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*104·4·2.*52·3.\supset \\ \vdash :\text{Hp}.\text{Hp}*104·4.&\supset .(\exists x,y,z).x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2.\\ &\iotaʻʻ\beta \in \text{N}^{1}\text{c}ʻ\beta \cap \text{Cl}ʻ1.\\ [*13·22] &\supset .(\exists x,y,z,\gamma ,\delta ).\gamma =x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz.\delta =\iotaʻʻ\beta .\\ &\gamma \in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2.\delta \in \text{N}^{1}\text{c}ʻ\beta \cap \text{Cl}ʻ1.\\ [*11·55] &\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2.\delta \in \text{N}^{1}\text{c}ʻ\beta \cap \text{Cl}ʻ1 &\qquad \text{(1)}\\ \vdash .(1).*101·35.\supset \vdash .\text{Prop} \end{array} \]

[Pg 50]

This proposition proves the desired conclusions provided \(\exists!\alpha\), and \(t_{0}ʻ\alpha\) consists of at least three terms. The following propositions deal with the cases in which this hypothesis is not verified.

*104·411. \(\vdash :tʻ\alpha =tʻ\beta .\alpha \in 0.\gamma =\Lambda _{\alpha }.\delta =\iotaʻʻ\beta .\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*73·47. &\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\alpha &\qquad \text{(1)}\\ *22·43.(*65·01). &\supset \vdash :\text{Hp}.\supset .\gamma \subset tʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*104·101.&\supset \vdash :\text{Hp}.\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(3)}\\ \vdash .(3).*104·2.*24·23.\supset \vdash .\text{Prop} \end{array} \]

*104·412. \[\begin{aligned}\vdash :tʻ\alpha =tʻ\beta .\alpha =\iota ʻx.\gamma =\iota ʻ\Lambda _{x}.&\delta =\iotaʻʻ\beta .\supset .\\ &\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*73·43. &\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\alpha &\qquad \text{(1)}\\ \vdash .*63·61·103. &\supset \vdash :\text{Hp}.\supset .\alpha \in t^{2}ʻx &\qquad \text{(2)}\\ \vdash .*22·43.(*65·01). \supset \vdash :\text{Hp}.\xi \in \gamma .&\supset _{\xi }.\xi \subset tʻx.\\ [*63·5] &\supset _{\xi }.\xi \in t^{2}ʻx.\\ [(2).*63·13] &\supset _\xi .\xi \in tʻ\alpha . &\qquad \text{(3)}\\ \vdash .(1).(3).*104·101.&\supset \vdash :\text{Hp}.\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(4)}\\ \vdash .*101·23. &\supset \vdash :\text{Hp}.\supset .\gamma \cap \delta =\Lambda &\qquad \text{(5)}\\ \vdash .(4).(5).*104·2.\supset \vdash .\text{Prop} \end{array} \]

*104·413. \[\begin{aligned}\vdash :tʻ\alpha =tʻ\beta .\alpha =\iota ʻx\cup \iota ʻy.&x \neq y.\gamma =\iota ʻ\Lambda \cup \iota ʻ(\iota ʻx\cup \iota ʻy).\delta =\iotaʻʻ\beta .\supset .\\ &\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*54·26. \supset \vdash :\text{Hp}.&\supset .\iota ʻx\cup \iota ʻy\in 2. &\qquad \text{(1)}\\ [*101·35] &\supset .\Lambda \neq \iota ʻx\cup \iota ʻy.\\ [*54·26] &\supset .\iota ʻ\Lambda \cup \iota ʻ(\iota ʻx\cup \iota ʻy)\in 2.\\ [*101·3] &\supset .\iota ʻ\Lambda \cup \iota ʻ(\iota ʻx\cup \iota ʻy)\in \text{Nc}ʻ(\iota ʻx\cup \iota ʻy) &\qquad \text{(2)}\\ \vdash .*51·16. \supset \vdash :\text{Hp}.&\supset .\alpha \in \gamma .\\ [*63·5] &\supset .\gamma \subset tʻ\alpha &\qquad \text{(3)}\\ \vdash .(2).(3).*104·1. &\supset \vdash :\text{Hp}.\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(4)}\\ \vdash .*52·21·3. &\supset \vdash .\Lambda {\sim}\in \iotaʻʻ\beta &\qquad \text{(5)}\\ \vdash .(1).*52·3.*54·25. &\supset \vdash :\text{Hp}.\supset .\iota ʻx\cup \iota ʻy{\sim}\in \iotaʻʻ\beta &\qquad \text{(6)}\\ \vdash .(5).(6). &\supset \vdash :\text{Hp}.\supset .\gamma \cap \delta =\Lambda &\qquad \text{(7)}\\ \vdash .(4).(7).*104·2.\supset \vdash .\text{Prop} \end{array} \]

*104·42. \[\begin{aligned}&\vdash :tʻ\alpha =tʻ\beta .\alpha \in 0\cup 1\cup 2.\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda \\ &[*104·411·412·413.*52·1.*54·101]\end{aligned}\]

[Pg 51]

*104·43. \(\vdash :tʻ\alpha =tʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda \)

Dem. \[ \begin{array}{l} \vdash .*54·56.\supset \\ \vdash :\text{Hp}.\alpha {\sim}\in 0\cup 1\cup 2.&\supset .(\exists x,y,z).x,y,z \in \alpha .x\neq y.x\neq z.y\neq z.\\ [*104·41] &\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda &\qquad \text{(1)}\\ \vdash .(1).*104·42.\supset \vdash .\text{Prop} \end{array} \]

The above proposition gives the desired result. The following propositions re-state this result in other forms.

*104·44. \[\begin{aligned}&\vdash :\mu ,\nu \in \text{N}^{1}\text{C}.tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu .\delta \in \nu .\gamma \cap \delta =\Lambda\\ &[*104·13·43]\end{aligned}\]

*104·45. \[\begin{aligned}&\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu ^{(1)}.\delta \in \nu ^{(1)}.\gamma \cap \delta =\Lambda\\ &[*104·29·44]\end{aligned}\]

*104·46. \[\begin{aligned}&\vdash :\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu ^{(1)}.\delta \in \nu ^{(1)}.\gamma \cap \delta =\Lambda \\ &[*104·28·44]\end{aligned}\]

Summary of *105.

In this number, we consider cardinals generated by a relation of similarity which goes from a higher to a lower type, i.e. given any class of classes \(\kappa\), we consider \(\text{Nc}ʻ\kappa\) in the type of members of \(\kappa\) (which we shall call \(\text{N}_{1}\text{c}ʻ\kappa\)) or in some lower type. Thus e.g. we shall have \[ \kappa =\iotaʻʻ\alpha .\supset .\alpha \in \text{N}_{1}\text{c}ʻ\kappa , \] where "\(\text{N}_{1}\text{c}ʻ\kappa\)" means "classes similar to \(\kappa\) but of the next lower type." Similarly \[ \kappa =\iotaʻʻ\iotaʻʻ\alpha .\supset .\alpha \in \text{N}_{2}\text{c}ʻ\kappa , \] and so on. We shall have generally \[ \begin{aligned} \beta \in \text{N}_{1}\text{c}ʻ\alpha .&\equiv \alpha \in \text{N}^{1}\text{c}ʻ\beta ,\\ \beta \in \text{N}_{2}\text{c}ʻ\alpha .&\equiv \alpha \in \text{N}^{2}\text{c}ʻ\beta , \end{aligned} \] and so on. The chief difference between ascending and descending cardinals is that \(\Lambda\) is one of the latter, but not one of the former. Otherwise the propositions of the present number are mostly analogous to corresponding propositions of *104.

On the analogy of the definitions in *104, we put \[ \begin{aligned} \text{N}_{1}\text{C}=\text{D}ʻ\text{N}_{1}\text{c} \quad\text{Df},\\ \mu _{(1)}=\text{ sm }ʻʻ\mu \cap t_{1}ʻ\mu \quad\text{Df}, \end{aligned} \] with similar definitions for \(\text{N}_{2}\text{C}\) and \(\mu_(2)\).

No proposition of the present number is ever referred to in the sequel, and the reader who is not interested in the subject may therefore omit it without detriment to what follows. The principal propositions proved are the following:

*105·25. \(\vdash .\text{N}_{0}\text{C}=\text{N}_{1}\text{C}-\iota ʻ\Lambda\)

*105·251. \(\vdash .\text{N}_{0}\text{C}=\text{N}_{2}\text{C}-\iota ʻ\Lambda\)

*105·26. \(\vdash .\text{N}_{1}\text{c}ʻtʻ\alpha =\Lambda\)

[Pg 53]

Thus \(\text{N}_{1}\text{C}\) or \(\text{N}_{2}\text{C}\), in any given type, only differs from \(\text{N}_{0}\text{C}\) in that type by the addition of \(\Lambda\).

*105·3. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha\)

*105·322. \(\vdash \colon\ldotp \exists !\text{N}_{1}\text{c}ʻ\alpha .\supset :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

*105·34. \(\vdash \colon\ldotp \mu \in \text{NC}.\exists !\mu _{(1)}.\supset :\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\equiv .\mu =\text{N}_{0}\text{c}ʻ\alpha\)

*105·35. \(\vdash \colon\ldotp \mu \in \text{NC}.\nu \in \text{N}_{0}\text{C}.\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu\)

*105·38. \(\vdash .\{\mu _{(1)}\}_{(1)}=\mu _{(2)}\)

*105·01. \(\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt_{1}ʻ\alpha \quad\text{Df}\)

We might write \[ \text{N}_{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha \quad\text{Df}, \] which would be equivalent to the above. But we choose the above form for the sake of uniformity. If \(s\) is any suffix, we put, provided \(t_{s}ʻ\alpha\) has been defined, \[ \text{N}_{s}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt_{s}ʻ\alpha \quad\text{Df}, \] and if \(i\) is any index for which \(t^{i}ʻ\alpha\) has been defined, we put \[ \text{N}^{i}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt^{i}ʻ\alpha \quad\text{Df}. \]

Thus for the sake of uniformity it is better, in the above definition *105·01, to write "\(tʻt_{1}ʻ\alpha\)" rather than "\(t_{0}ʻ\alpha\)."

*105·011. \(\text{N}_{2}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt_{2}ʻ\alpha \quad\text{Df}\)

*105·02. \(\text{N}_{1}\text{C}=\text{D}ʻ\text{N}_{1}\text{c} \quad\text{Df}\)

*105·021. \(\text{N}_{2}\text{C}=\text{D}ʻ\text{N}_{2}\text{c} \quad\text{Df}\)

*105·03. \(\mu _{(1)}=\text{ sm }ʻʻ\mu \cap t_{1}ʻ\mu \quad\text{Df}\)

*105·031. \(\mu _{(2)}=\text{ sm }ʻʻ\mu \cap t_{2}ʻ\mu \quad\text{Df}\)

*105·1. \(\vdash .\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha \quad[*63·383.(*105·01)]\)

*105·101. \(\vdash .\text{N}_{2}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{1}ʻ\alpha \quad[*63·41.(*105·011)]\)

*105·11. \[\begin{aligned}&\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in t_{0}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \subset t_{1}ʻ\alpha \\ &[*105·1.*100·31.*63·51]\end{aligned}\]

*105·111. \[\begin{aligned}&\vdash :\beta \in \text{N}_{2}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in t_{1}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \in t_{1}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \subset t_{2}ʻ\alpha\\ &[*105·101.*100·31.*63·52]\end{aligned}\]

*105·12. \[\begin{aligned}&\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\alpha \subset tʻ\beta .\equiv .\beta \text{ sm }\alpha .\alpha \subset tʻ\beta .\equiv .\alpha \in \text{N}^{1}\text{c}ʻ\beta \\ &[*105·11.*63·51.*104·1]\end{aligned}\]

*105·121. \[\begin{aligned}&\vdash :\beta \in \text{N}_{2}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\alpha \subset t^{2}ʻ\beta .\equiv .\beta \text{ sm }\alpha .\alpha \subset t^{2}ʻ\beta .\equiv .\alpha \in \text{N}^{2}\text{c}ʻ\beta \\ &[*105·111.*63·52.*104·11]\end{aligned}\]

*105·13. \(\vdash .\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}(t_{1}ʻ\alpha )ʻ\alpha =\text{Nc}\{(t_{1}ʻ\alpha )_{\alpha }\}ʻ\alpha \quad[*102·6.(*105·.01)]\)

[Pg 54]

*105·131. \(\vdash .\text{N}_{2}\text{c}ʻ\alpha =\text{Nc}(t_{2}ʻ\alpha )ʻ\alpha =\text{Nc}\{(t_{2}ʻ\alpha )_{\alpha }\}ʻ\alpha \quad[*102·6.(*105·011)]\)

*105·14. \(\vdash :\alpha \in t_{0}ʻ\beta .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{Nc}(a)ʻ\beta =\text{Nc}(a_{\beta })ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*63·22.\supset \vdash :\text{Hp}.&\supset .tʻ\alpha =t_{0}ʻ\beta .\\ [*105·1] &\supset .\text{N}_{1}\text{c}ʻ\beta =\text{Nc}ʻ\beta \cap tʻ\alpha &\qquad \text{(1)}\\ \vdash .(1).*102·6.\supset \vdash .\text{Prop} \end{array} \]

*105·141. \(\vdash :\alpha \in t_{1}ʻ\beta .\supset .\text{N}_{2}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\beta })ʻ\beta \quad[\text{Proof as in}\, *105·14]\)

*105·142. \(\vdash :\beta \subset tʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\beta })ʻ\beta \quad[*105·14.*63·51]\)

*105·143. \(\vdash :\beta \subset t^{2}ʻ\alpha .\supset .\text{N}_{2}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\beta })ʻ\beta \quad[*105·141.*63·52]\)

*105·15. \(\vdash :\mu \in \text{N}_{1}\text{C}.\equiv .(\exists \alpha ).\mu =\text{N}_{1}\text{c}ʻ\alpha \quad[*100·22.*71·41.(*105·02)]\)

*105·151. \(\vdash :\mu \in \text{N}_{2}\text{C}.\equiv .(\exists \alpha ).\mu =\text{N}_{2}\text{c}ʻ\alpha\)

*105·16. \[\begin{aligned}\vdash :\delta \in \mu _{(1)}.&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{1}ʻ\mu .\\ &\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{0}ʻ\gamma .\\ &\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\gamma \subset tʻ\delta \quad[*37·1.*63·51·54]\end{aligned}\]

*105·161. \[\begin{aligned}\vdash :\delta \in \mu _{(2)}.&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{2}ʻ\mu .\\ &\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{1}ʻ\gamma .\\ &\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\gamma \subset t^{2}ʻ\delta \quad[*37·1.*63·52·55]\end{aligned}\]

In what follows, propositions concerning \(\text{N}_{2}\text{c}\) or \(\text{N}_{2}\text{C}\) have proofs exactly analogous to those of the corresponding propositions concerning \(\text{N}_{1}\text{c}\) or \(\text{N}_{1}\text{C}\).

*105·2. \(\vdash .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*105·12.*104·2.&\supset \vdash .\alpha \in \text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha .\\ [*103·26]&\supset \vdash .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha \end{array} \]

*105·201. \(\vdash .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha\)

*105·21. \(\vdash .\text{N}_{0}\text{C}\subset \text{N}_{1}\text{C} \quad[*105·2·15]\)

*105·211. \(\vdash .\text{N}_{0}\text{C}\subset \text{N}_{2}\text{C}\)

*105·22. \(\vdash :\gamma \in \text{N}_{1}\text{c}ʻ\delta .\supset .\text{N}_{1}\text{c}ʻ\delta =\text{N}_{0}\text{c}ʻ\gamma \quad[*103·26]\)

*105·221. \(\vdash :\gamma \in \text{N}_{2}\text{c}ʻ\delta .\supset .\text{N}_{2}\text{c}ʻ\delta =\text{N}_{0}\text{c}ʻ\gamma\)

*105·23. \(\vdash :\exists !\text{N}_{1}\text{c}ʻ\delta .\supset .\text{N}_{1}\text{c}ʻ\delta \in \text{N}_{0}\text{C} \quad[*105·22]\)

*105·231. \(\vdash :\exists !\text{N}_{2}\text{c}ʻ\delta .\supset .\text{N}_{2}\text{c}ʻ\delta \in \text{N}_{0}\text{C}\)

*105·24. \(\vdash .\text{N}_{1}\text{C}-\iota ʻ\Lambda \subset \text{N}_{0}\text{C} \quad[*105·23]\)

*105·241. \(\vdash .\text{N}_{2}\text{C}-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}\)

*105·25. \(\vdash .\text{N}_{0}\text{C}=\text{N}_{1}\text{C}-\iota ʻ\Lambda \quad[*105·21·24.*103·23]\)

[Pg 55]

*105·251. \(\vdash .\text{N}_{0}\text{C}=\text{N}_{2}\text{C}-\iota ʻ\Lambda\)

*105·252. \(\vdash .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\iotaʻʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*105·111.\supset \vdash :\alpha \in \text{N}_{2}\text{c}ʻ\iotaʻʻ\beta .&\equiv .\alpha \text{ sm }\iotaʻʻ\beta .\alpha \in t_{1}ʻ\iotaʻʻ\beta .\\ [*73·41.*63·64·54] &\equiv .\alpha \text{ sm }\beta .\alpha \in t_{0}ʻ\beta .\\ [*105·11] &\equiv .\alpha \in \text{N}_{1}\text{c}ʻ\beta :\supset \vdash .\text{Prop} \end{array} \]

*105·26. \(\vdash .\text{N}_{1}\text{c}ʻtʻ\alpha =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*105·142.\supset \vdash .\text{N}_{1}\text{c}ʻtʻ\alpha =\text{Nc}(\alpha )ʻtʻ\alpha &\qquad \text{(1)}\\ \vdash .(1).*102·73.\supset \vdash .\text{Prop} \end{array} \]

*105·261. \(\vdash .\text{N}_{2}\text{c}ʻ\iotaʻʻtʻ\alpha =\Lambda \quad[*105·26·252]\)

*105·27. \(\vdash .\Lambda \in \text{N}_{1}\text{C} \quad[*105·26]\)

*105·271. \(\vdash .\Lambda \in \text{N}_{2}\text{C}\)

*105·28. \(\vdash .\text{N}_{1}\text{C}=\text{N}_{0}\text{C}\cup \iota ʻ\Lambda \quad[*105·25·27]\)

*105·281. \(\vdash .\text{N}_{2}\text{C}=\text{N}_{1}\text{C}=\text{N}_{0}\text{C}\cup \iota ʻ\Lambda\)

*105·29. \(\vdash .\text{NC}\subset \text{N}_{1}\text{C}.\text{NC}\subset \text{N}_{2}\text{C} \quad[*105·281.*103·34]\)

*105·3. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*103·4.(*105·03).\supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .&\supset .\mu _{(1)}=\text{Nc}ʻ\alpha \cap t_{1}ʻ\mu &\qquad \text{(1)}\\ \vdash .*103·12. &\supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\alpha \in \mu .\\ [*63·105] &\supset .\alpha \in t_{0}ʻ\mu .\\ [*63·54] &\supset .t_{0}ʻ\alpha =t_{1}ʻ\mu &\qquad \text{(2)}\\ \vdash .(1).(2). \supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .&\supset .\mu _{(1)}=\text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha .\\ [*105·1] &\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*105·301. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha\)

*105·31. \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu {(1)}\in \text{N}_{1}\text{C} \quad[*105·3·15.*103·2]\)

*105·311. \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu _{(2)}\in \text{N}_{2}\text{C}\)

*105·312. \(\vdash :\gamma \in \text{N}_{1}\text{c}ʻ\alpha .\supset .\alpha \in \text{N}^{1}\text{c}ʻ\gamma .\text{N}^{1}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha \quad[*105·12.*103·26]\)

*105·313. \(\vdash :\gamma \in \text{N}_{2}\text{c}ʻ\alpha .\supset .\alpha \in \text{N}^{2}\text{c}ʻ\gamma .\text{N}^{2}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha\)

*105·314. \(\vdash :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma \quad[*105·312.*103·12]\)

*105·315. \(\vdash :\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{2}\text{c}ʻ\gamma\)

*105·316. \(\vdash :\exists !\text{N}_{1}\text{c}ʻ\alpha .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*105·312.&\supset \vdash :\gamma \in \text{N}_{1}\text{c}ʻ\alpha \\.\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .&\supset .\text{N}^{1}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha .\text{N}^{1}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\beta .\\ [*13·171] &\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*10·11·23·35.\supset \vdash .\text{Prop} \end{array} \]

[Pg 56]

*105·317. \(\vdash :\exists !\text{N}_{2}\text{c}ʻ\alpha .\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

*105·32. \(\vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*103·41.\supset \vdash :\text{Hp}.&\supset .\text{Nc}(t_{1}ʻ\alpha )ʻ\alpha =\text{Nc}(t_{1}ʻ\alpha )ʻ\beta &\qquad \text{(1)}\\ \vdash .*103·14.\supset \vdash :\text{Hp}.&\supset .\beta \in tʻ\alpha .\\ [*63·16·36] &\supset .t_{1}ʻ\alpha =t_{1}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash :\text{Hp}.&\supset .\text{Nc}(t_{1}ʻ\alpha )ʻ\alpha =\text{Nc}(t_{1}ʻ\beta )ʻ\beta .\\ [*105·13] &\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta :\supset \vdash .\text{Prop} \end{array} \]

*105·321. \(\vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\beta\)

*105·322. \(\vdash \colon\ldotp \exists !\text{N}_{1}\text{c}ʻ\alpha .\supset :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta \quad[*105·316·32]\)

*105·323. \(\vdash \colon\ldotp \exists !\text{N}_{2}\text{c}ʻ\alpha .\supset :\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)

*105·324. \(\vdash :\exists !\mu _{(1)}.\supset .\exists !\mu \quad[*37·29.(*105·03)]\)

*105·325. \(\vdash :\exists !\mu _{(2)}.\supset .\exists !\mu \)

*105·326. \(\vdash :\mu \in \text{NC}.\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma .\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash .*103·26. \supset \vdash :\text{Hp}.\alpha \in \mu .&\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha . &\qquad \text{(1)}\\ [*105·3] & \supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\\ [\text{Hp}] &\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\\ [*105·314] & \supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma .\\ [(1)] &\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma &\qquad \text{(2)}\\ \vdash .(2).*10·11·23·35.\supset \vdash :\text{Hp}.\exists !\mu .&\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma &\qquad \text{(3)}\\ \vdash .(3).*105·324.*103·13.\supset \vdash .\text{Prop} \end{array} \]

*105·327. \(\vdash :\mu \in \text{NC}.\mu _{(2)}=\text{N}_{0}\text{c}ʻ\gamma .\supset .\mu =\text{N}^{2}\text{c}ʻ\gamma\)

*105·33. \(\vdash :\mu \in \text{NC}.\exists !\mu _{(1)}.\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*103·26.\supset \vdash :\gamma \in \mu _{(1)}.\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\\ [*105·314] \supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma &\qquad \text{(1)}\\ \vdash .(1).*105·326.\supset \\ \vdash :\gamma \in \mu _{(1)}.\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\mu \in \text{NC}.\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha &\qquad \text{(2)}\\ \vdash .(2).*10·11·23·35.\supset \vdash .\text{Prop} \end{array} \]

*105·331. \(\vdash :\mu \in \text{NC}.\exists !\mu _{(2)}.\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha\)

*105·34. \(\vdash \colon\ldotp \mu \in \text{NC}.\exists !\mu _{(1)}.\supset :\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\equiv .\mu =\text{N}_{0}\text{c}ʻ\alpha \quad[*105·33·3]\)

*105·341. \(\vdash \colon\ldotp \mu \in \text{NC}.\exists !\mu _{(2)}.\supset :\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha .\equiv .\mu =\text{N}_{0}\text{c}ʻ\alpha\)

*105·342. \(\vdash .\mu \in \text{NC}.\supset .\mu _{(1)}\in \text{N}_{1}\text{C}\)

Dem. \[ \begin{array}{l} \vdash .*103·34. \supset \vdash :\text{Hp}.\exists !\mu .&\supset .\mu \in \text{N}_{0}\text{C}.\\ [*105·31] &\supset .\mu _{(1)}\in \text{N}_{1}\text{C} &\qquad \text{(1)}\\ \vdash .*105·324.\supset \vdash :\text{Hp}.{\sim}\exists !\mu .&\supset .{\sim}\exists !\mu _{(1)}.\\ [*105·27] &\supset .\mu _{(1)}\in \text{N}_{1}\text{C} &\qquad \text{(2)}\\ \vdash .(1).(2). \supset \vdash .\text{Prop} \end{array} \]

[Pg 57]

*105·343. \(\vdash :\mu \in \text{NC}.\supset .\mu _{(2)}\in \text{N}_{2}\text{C}\)

*105·344. \(\vdash :\mu =\text{N}^{1}\text{c}ʻ\gamma .\supset .\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *104·24.\supset \vdash :\text{Hp}.&\supset .\mu =\text{N}_{0}\text{c}ʻ\iotaʻʻ\gamma .\\ [*105·3] &\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\iotaʻʻ\gamma .\\ [*105·2] &\supset .\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma :\supset \vdash .\text{Prop} \end{array} \]

*105·345. \(\vdash :\mu =\text{N}^{2}\text{c}ʻ\gamma .\supset .\mu _{(2)}=\text{N}_{0}\text{c}ʻ\gamma\)

*105·35. \(\vdash \colon\ldotp \mu \in \text{NC}.\nu \in \text{N}_{0}\text{C}.\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu\)

Dem. \[ \begin{array}{l} \vdash .*105·326.*104·26.\supset \\ \vdash :\mu \in \text{NC}.\nu =\text{N}_{0}\text{c}ʻ\gamma .\mu _{(1)}=\nu .\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma . \nu ^{(1)}=\text{N}_{1}\text{c}ʻ\gamma .\\ [*13·172] \supset .\mu =\nu ^{(1)} &\qquad \text{(1)}\\ \vdash .*104·26.\text{Fact}.\supset \\ \vdash :\mu \in \text{NC}.\nu =\text{N}_{0}\text{c}ʻ\gamma .\mu =\nu ^{(1)}.\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\gamma .\\ [*105·344] \supset .\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\gamma .\\ [*13·172] \supset .\mu _{(1)}=\nu &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash \colon\ldotp \mu \in \text{NC}.\nu =\text{N}_{0}\text{c}ʻ\gamma .\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu &\qquad \text{(3)}\\ \vdash .(3).*103·2.\supset \vdash .\text{Prop} \end{array} \]

*105·351. \(\vdash \colon\ldotp \mu \in \text{NC}.\nu \in \text{N}_{0}\text{C}.\supset :\mu =\nu ^{(2)}.\equiv .\mu _{(2)}=\nu\)

*105·352. \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\exists !\nu .\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu \quad[*105·35.*103·34]\)

*105·353. \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\exists !\nu .\supset :\mu =\nu ^{(2)}.\equiv .\mu _{(2)}=\nu\)

*105·354. \(\vdash :\nu \in \text{NC}.\exists !\nu .\supset .{\nu ^{(1)}}_{(1)}=\nu \quad[*105·352]\)

*105·355. \(\vdash :\nu \in \text{NC}.\exists !\nu .\supset .\{\nu ^{(2)}\}_{(2)}=\nu\)

*105·356. \(\vdash :\mu \in \text{NC}.\exists !\mu _{(1)}.\supset .\{\mu _{(1)}\}^{(1)}=\mu \quad[*105·352]\)

*105·357. \(\vdash :\mu \in \text{NC}.\exists !\mu _{(2)}.\supset .\{\mu _{(2)}\}^{(2)}=\mu\)

*105·36. \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\gamma \in \text{N}_{1}\text{c}ʻ\beta .\supset .\gamma \in \text{N}_{2}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*105·11.\supset \vdash :\text{Hp}.&\supset .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta .\\ [*73·32.*63·38] &\supset .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\\ [*105·111] &\supset .\gamma \in \text{N}_{2}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*105·361. \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\gamma \in \text{N}_{2}\text{c}ʻ\alpha .\supset .\gamma \in \text{N}_{1}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*105·11·111 .\supset \vdash :\text{Hp}.&\supset .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\\ [*73·31·32] &\supset .\gamma \text{ sm }\beta .\beta \in t_{0}ʻ\alpha .\gamma \in t_{1}ʻ\alpha &\qquad \text{(1)}\\ \vdash .*63·54. &\supset \vdash :\beta \in t_{0}ʻ\alpha .\supset .t_{0}ʻ\beta =t_{1}ʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta .\\ [*105·11] \supset .\gamma \in \text{N}_{1}\text{c}ʻ\beta :\supset \vdash .\text{Prop} \end{array} \]

[Pg 58]

*105·362. \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\alpha \quad[*105·36·361]\)

*105·37. \(\vdash :\text{N}_{0}\text{c}ʻ\beta =\text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\alpha \quad[*105·362.*103·12]\)

*105·371. \(\vdash :\exists !\mu _{(2)}.\supset .\exists !\mu _{(1)}\)

Dem. \[ \begin{array}{l} \vdash .*63·381.(*63·05).\supset \\ \vdash :\gamma \text{ sm }\alpha .\alpha \in \mu .\gamma \in t_{2}ʻ\mu .&\supset .\gamma \text{ sm }\alpha .\alpha \in \mu .tʻ\gamma =t_{2}ʻ\mu .\\ [*73·41.*63·64] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\alpha \in \mu .t_{0}ʻ\iotaʻʻ\gamma =t_{2}ʻ\mu .\\ [*63·57] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\alpha \in \mu .tʻ\iotaʻʻ\gamma =t_{1}ʻ\mu .\\ [*63·103] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\alpha \in \mu .\iotaʻʻ\gamma \in t_{1}ʻ\mu .\\ [*105·16] &\supset .\iotaʻʻ\gamma \in \mu _{(1)}.\\ [*10·24] &\supset .\exists !\mu _{(1)} &\qquad \text{(1)}\\ \vdash .(1).*10·11·23.\supset \\ \vdash :(\exists \alpha ).\gamma \text{ sm }\alpha .\alpha \in \mu .\gamma \in t_{2}ʻ\mu .&\supset .\exists !\mu _{(1)} &\qquad \text{(2)}\\ \vdash .(2).*105·161.\supset \vdash :\gamma \in \mu _{(2)}.&\supset .\exists !\mu _{(1)} &\qquad \text{(3)}\\ \vdash .(3).*10·11·23.\supset \vdash .\text{Prop} \end{array} \]

*105·372. \(\vdash :\mu _{(1)}=\Lambda .\supset .\mu _{(2)}=\Lambda \quad[*105·371.\text{Transp}]\)

*105·38. \(\vdash .\{\mu _{(1)}\}_{(1)}=\mu _{(2)}\)

Dem. \[ \begin{array}{l} \vdash .*105·16.\supset \vdash :\gamma \in \{\mu _{(1)}\}_{(1)}.&\equiv .(\exists \beta ).\beta \in \mu _{(1)}.\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta .\\ [*105·16] &\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta . &\qquad \text{(1)}\\ [*73·32.*63·38] &\supset .(\exists \alpha ).\alpha \in \mu .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha &\qquad \text{(2)}\\ \vdash .*73·41.*63·64·53·57.\supset \\ \vdash :\alpha \in \mu .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .&\supset .\alpha \in \mu .\iotaʻʻ\gamma \text{ sm }\alpha .\gamma \text{ sm }\iotaʻʻ\gamma .\gamma \in t_{0}ʻ\iotaʻʻ\gamma .\iotaʻʻ\gamma \in t_{0}ʻ\alpha .\\ [(1)] &\supset .\gamma \in \{\mu _{(1)}\}_{(1)} &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash :\gamma \in \{\mu _{(1)}\}_{(1)}.&\equiv .(\exists \alpha ).\alpha \in \mu .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\\ [*105·161] &\equiv .\gamma \in \mu _{(2)}:\supset \vdash .\text{Prop} \end{array} \]

*105·4. \(\vdash :\gamma \in \text{N}_{2}\text{c}ʻ\alpha .\supset .\iotaʻʻ\gamma \in \text{N}_{1}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*105·111.*73·41.*63·64.\supset \vdash :\text{Hp}.&\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\gamma \in t_{0}ʻ\iotaʻʻ\gamma .\\ [*63·41·383·16·55] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .t_{1}ʻ\alpha =t_{0}ʻ\iotaʻʻ\gamma .\\ [*63·54] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\iotaʻʻ\gamma \in t_{0}ʻ\alpha .\\ [*105·11] &\supset .\iotaʻʻ\gamma \in \text{N}_{1}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*105·41. \(\vdash :\exists !\text{N}_{2}\text{c}ʻ\alpha .\supset .\exists !\text{N}_{1}\text{c}ʻ\alpha \quad[*105·4]\)

[Pg 59]

*105·42. \(\vdash :\text{N}_{1}\text{c}ʻ\alpha =\Lambda .\supset .\text{N}_{2}\text{c}ʻ\alpha =\Lambda \quad[*105·41]\)

*105·43. \(\vdash :\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\supset .\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha \)

Dem. \[ \begin{array}{l} \vdash .*105·11. \supset \vdash :\text{Hp}.\beta \in \mu _{(1)}.&\supset .\beta \in \text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha .\\ [*63·54.*100·31·321] &\supset .\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha .t_{0}ʻ\beta =t_{1}ʻ\alpha .\\ [*105·1·101] &\supset .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\alpha &\qquad \text{(1)}\\ \vdash .*105·3.*103·26.\supset \vdash :\text{Hp}.\beta \in \mu _{(1)}.\supset .\text{N}_{1}\text{c}ʻ\beta &=\{\mu _{(1)}\}_{(1)}\\ [*105·38]&=\mu _{(2)} &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash :\text{Hp}.\exists !\mu _{(1)}.\supset .\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha &\qquad \text{(3)}\\ \vdash .*105·372·42. &\supset \vdash :\text{Hp}.\mu _{(1)}=\Lambda .\supset .\mu _{(2)}=\Lambda .\text{N}_{2}\text{c}ʻ\alpha =\Lambda &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*105·44. \(\vdash .\text{N}_{2}\text{c}ʻt^{2}ʻ\alpha =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*105·26.&\supset \vdash .\text{N}_{1}\text{c}ʻtʻtʻ\alpha =\Lambda .\\ [*105·42] &\supset \vdash .\text{N}_{2}\text{c}ʻtʻtʻ\alpha =\Lambda .\supset \vdash .\text{Prop} \end{array} \]

Summary of *106.

In this number we have to consider the cardinals whose members are classes of relations which have a given relation of type to some given class. For example, we have \(\downarrow xʻʻ\alpha \text{ sm }\alpha\), and \(\downarrow xʻʻ\alpha\) has a given relation of type to \(\alpha\) when \(x\) is given. Thus we want a notation for \[ \text{Nc}ʻ\alpha \cap tʻ\downarrow xʻʻ\alpha \] and all the associated ideas. In this number, we shall deal only with relations in which the referent and relatum have a relation, as to type, which can be expressed by the notations of *63, i.e. roughly speaking, when, for suitable values of \(\alpha\), \(m\), \(n\), our relations are contained in \[ t^{m}ʻ\alpha \uparrow t^{n}ʻ\alpha\,\,\, \text{or}\,\,\, t_{m}ʻ\alpha \uparrow t_{n}ʻ\alpha\,\,\, \text{or}\,\,\, t^{m}ʻ\alpha \uparrow t_{n}ʻ\alpha\,\,\, \text{or}\,\,\, t_{m}ʻ\alpha \uparrow t^{n}ʻ\alpha. \]

Thus if \(t_{\mu \nu }ʻ\alpha\) has been defined, we shall put \[ \begin{aligned} \text{N}_{\mu \nu }\text{c}ʻ\alpha &=\text{Nc}ʻ\alpha \cap tʻt_{\mu \nu }ʻ\alpha &\quad\text{Df},\\ \text{N}_{\mu \nu }\text{C}&=\text{D}ʻ\text{N}_{\mu \nu }\text{c} &\quad\text{Df},\\ \xi _{\mu \nu }&=\text{ sm }ʻʻ\xi \cap tʻt_{\mu \nu }ʻt_{1}ʻ\xi &\quad\text{Df}, \end{aligned} \] with analogous definitions for \(t^{\mu \nu }ʻ\alpha\), \(t^\mu _{\nu}ʻ\alpha\) and \(^\mu t_{\nu }ʻ\alpha\).

Much the most important case is that of \(t_{00}ʻ\alpha\). For this case we have

*106·1. \[\begin{aligned}\vdash :\beta \in \text{N}_{00}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt_{00}ʻ\alpha .&\equiv .\beta \text{ sm }\alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha ).\\ &\equiv .\beta \text{ sm }\alpha .\beta \subset tʻ(\alpha \uparrow \alpha )\end{aligned}\]

Thus \(\text{N}_{00}\text{c}ʻ\alpha\) will be the number of a class of relations whose fields are of the same type as \(\alpha\), provided this class of relations is similar to \(\alpha\). E.g. the number of terms such as \(x\downarrow x\), where \(x\in \alpha\), will be \(\text{N}_{00}\text{c}ʻ\alpha\).

We have

*106·21. \(\vdash .\exists !\text{N}_{00}\text{c}ʻ\alpha .\text{N}_{00}\text{c}ʻ\alpha \in \text{N}_{0}\text{C}\)

*106·22.\(\vdash :\lambda \in \text{N}_{0}^{1}\text{c}ʻ\alpha .\equiv .\text{Cnv}ʻʻ\lambda \in ^{1}\text{N}_{0}\text{c}ʻ\alpha \)

*106·23. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta\)

*106·32. \(\vdash :t_{0}ʻ\alpha =t_{0}ʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\)

[Pg 61]

*106·4·41·411. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(00)}=\text{N}_{00}\text{c}ʻ\alpha .\mu ^{(11)}=\text{N}^{11}\text{c}ʻ\alpha .\mu _{(11)}=\text{N}_{11}\text{c}ʻ\alpha\)

*106·53. \(\vdash .\text{Nc}(\alpha )ʻt_{00}ʻ\alpha = \Lambda\)

whence it follows that

*106·54. \(\vdash .\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha {\sim} \in \text{N}_{00}\text{C}\)

The propositions of this number, except *106·21, are never referred to again (except in *154·25·251·262, which are themselves never used again), but they have a somewhat greater importance than the propositions of *105, owing to the fact that the arithmetical operations are defined by means of classes of relations, i.e. the sum of two cardinals (for instance) is defined as the cardinal number of a certain class of relations (cf. *110).

*106·01 \(\text{N}_{00}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt_{00}ʻ\alpha \quad\text{Df}\)

*106·011. \(\text{N}^{11}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt^{11}ʻ\alpha \quad\text{Df}\)

*106·012. \(\text{N}_{01}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt_{01}ʻ\alpha \quad\text{Df etc.}\)

*106·02. \(\text{N}^{1}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt_{0}^{1}ʻ\alpha \quad\text{Df etc.}\)

*106·021. \(^{1}\text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻ^{1}t_{0}ʻ\alpha \quad\text{Df etc.}\)

*106·03. \(\text{N}_{00}\text{C} = \text{D}ʻ\text{N}_{00}\text{c} \quad\text{Df etc.}\)

*106·04. \(\mu _{(00)} = \text{ sm }ʻʻ\mu \cap tʻt_{00}ʻt_{1}ʻ\mu \quad\text{Df}\)

*106·041. \(\mu ^{(11)} = \text{ sm }ʻʻ\mu \cap tʻt^{11}ʻt_{1}ʻ\mu \quad\text{Df etc.}\)

*106·1. \[\begin{aligned}\vdash :\beta \in \text{N}_{00}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt_{00}ʻ\alpha .\\ &\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha ).\\ &\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(\alpha \uparrow \alpha )\\ &[*100·1.(*106·01.*64·01).*64·11]\end{aligned}\]

*106·101. \[\begin{aligned}\vdash :\beta \in \text{N}^{11}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt^{11}ʻ\alpha .\\ &\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\alpha ).\\ &\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(tʻ\alpha \uparrow tʻ\alpha )\end{aligned}\]

Similar propositions hold for any other double index \(mn\) for which \(t^{mn}ʻ\alpha\) has been defined.

*106·11. \[\begin{aligned}\vdash :\beta \in \text{N}_{01}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta tʻt_{01}ʻ\alpha .\\ &\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow t_{1}ʻ\alpha ).\\ &\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(t_{0}ʻ\alpha \uparrow t_{1}ʻ\alpha )\end{aligned}\]

Similar propositions hold for any other double suffix \(mn\) for which \(t_{mn}ʻ\alpha\) has been defined.

*106·12. \[\begin{aligned}\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt_{0}^{1}ʻ\alpha .\\ &\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow tʻ\alpha ).\\ &\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(t_{0}ʻ\alpha \uparrow tʻ\alpha )\end{aligned}\]

*106·121. \[\begin{aligned}\vdash :\beta \in ^{1}\text{N}_{0}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻ^{1}t_{0}ʻ\alpha .\\ &\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(tʻ\alpha \uparrow t_{0}ʻ\alpha ).\\ &\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(tʻ\alpha \uparrow t_{0}ʻ\alpha )\end{aligned}\]

[Pg 62]

Similar propositions hold for any other index and suffix for which \(t_{m}^{n}ʻ\alpha\) or \(^{n}t_{m}ʻ\alpha\) has been defined.

*106·13. \(\vdash :\mu \in \text{N}_{00}\text{C}.\equiv .(\exists \alpha ).\mu =\text{N}_{00}\text{c}ʻ\alpha \quad[*100·22.*71·41]\)

Similar propositions hold for \(\text{N}^{11}\text{C}ʻ\alpha\) etc.

*106·14. \[\begin{aligned}\vdash :\beta \in \mu _{(00)}.&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻtʻ(t_{1}ʻ\mu \uparrow t_{1}ʻ\mu ).\\ &\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻt_{00}ʻ\alpha .\\ &\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \subset tʻ(\alpha \uparrow \alpha ) \quad[*64·33·11]\end{aligned}\]

*106·141. \[\begin{aligned}\vdash :\beta \in \mu _{0}^{1}.&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻtʻ(t_{1}ʻ\mu \uparrow t_{0}ʻ\mu ).\\ &\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻt_{0}^{1}ʻ\alpha .\\ &\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \subset tʻ(\alpha \uparrow tʻ\alpha )\end{aligned}\]

Similar propositions hold for \(^{1}\mu _{0}\), \(\mu ^{11}\), \(\mu_{11}\) etc.

*106·2 \(\vdash :x\in t_{0}ʻ\alpha .\supset .\downarrow xʻʻ\alpha \in \text{N}_{00}\text{c}ʻ\alpha .\downarrow xʻʻ\alpha \in \text{N}_{0}\text{c}ʻ\downarrow xʻʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*55·15.&\supset \vdash :R\in \downarrow xʻʻ\alpha .\supset .\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR=\iota ʻx:\\ [*63·105] &\supset \vdash \colon\ldotp x\in t_{0}ʻ\alpha .\supset :R\in \downarrow xʻʻ\alpha .\supset _{R}.\text{D}ʻR\subset t_{0}ʻ\alpha .\text{ᗡ}ʻR\subset t_{0}ʻ\alpha .\\ [*35·83] &\supset _{R}.R\unicode{x2abd} t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha .\\ [*64·16·13] &\supset _{R}.R\in tʻ(\alpha \uparrow \alpha ):\\ [*22·1] &\supset :\downarrow xʻʻ\alpha \subset tʻ(\alpha \uparrow \alpha ) &\qquad \text{(1)}\\ \vdash .(1).*73·611.*106·1.*103·12.&\supset \vdash .\text{Prop} \end{array} \]

*106·201. \(\vdash :\beta \in tʻ\alpha .\supset .\downarrow \beta ʻʻ\alpha \in \text{N}_{0}^{1}\text{c}ʻ\alpha \)

*106·202. \(\vdash :\beta \in t^{2}ʻ\alpha .\supset .\downarrow \beta ʻʻ\alpha \in \text{N}_{0}^{2}\text{c}ʻ\alpha\)

*106·203. \(\vdash .\downarrow \alpha ʻʻ\alpha \in \text{N}_{0}^{1}\text{c}ʻ\alpha \quad[*106·201]\)

*106·204. \(\vdash .\downarrow (\iotaʻʻ\alpha )ʻʻ\alpha \in \text{N}_{0}^{2}\text{c}ʻ\alpha \quad[*106·202]\)

*106·21. \(\vdash .\exists !\text{N}_{00}\text{c}ʻ\alpha .\text{N}_{00}\text{c}ʻ\alpha \in \text{N}_{0}\text{C} \quad[*106·2.*63·18]\)

*106·211. \(\vdash .\Lambda {\sim}\in \text{N}_{00}\text{C}.\text{N}_{00}\text{C}\subset \text{N}_{0}\text{C}.\text{N}_{00}\text{C}\in \text{Cls ex}^{2}\,\text{excl} \quad[*106·21.*103·24]\)

*106·212. \(\vdash .\Lambda {\sim}\in \text{N}_{0}^{1}\text{C}.\text{N}_{0}^{1}\text{C}\subset \text{N}_{0}\text{C}.\text{N}_{0}^{1}\text{C}\in \text{Cls ex}^{2}\,\text{excl} \quad[*106·203]\)

*106·213. \(\vdash .\Lambda {\sim}\in \text{N}_{0}^{2}\text{C}.\text{N}_{0}^{2}\text{C}\subset \text{N}_{0}\text{C}.\text{N}_{0}^{2}\text{C}\in \text{Cls ex}^{2}\,\text{excl} \quad[*106·204]\)

*106·22. \(\vdash :\lambda \in \text{N}_{0}^{1}\text{c}ʻ\alpha .\equiv .\text{Cnv}ʻʻ\lambda \in ^{1}\text{N}_{0}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*73·4. \supset \vdash :\lambda \text{ sm }\alpha .&\equiv .\text{Cnv}ʻʻ\lambda \text{ sm }\alpha &\qquad \text{(1)}\\ \vdash .*64·16. \supset \vdash \colon\ldotp \lambda \subset tʻ(t_{0}ʻ\alpha \uparrow tʻ\alpha ).&\equiv :R\in \lambda .\supset _{R}.R\,\unicode{x2abd}\,t_{0}ʻ\alpha \uparrow tʻ\alpha :\\ [*35·84] &\equiv :R\in \lambda .\supset _{R}.\breve{R}\,\unicode{x2abd}\, tʻ\alpha \uparrow t_{0}ʻ\alpha :\\ [*37·63] &\equiv :S\in \text{Cnv}ʻʻ\lambda .\supset _{S}.S\,\unicode{x2abd}\, tʻ\alpha \uparrow t_{0}ʻ\alpha :\\ [*64·16] &\equiv :\text{Cnv}ʻʻ\lambda \subset tʻ(tʻ\alpha \uparrow t_{0}ʻ\alpha ) &\qquad \text{(2)}\\ \vdash .(1).(2).*106·12.\supset \vdash .\text{Prop} \end{array} \]

[Pg 63]

The proof requires, in addition to *106·12, its analogue for \(^{1}\text{N}_{0}\text{c}ʻ\alpha\). Such analogues will be assumed as required.

*106·221. \(\vdash :\lambda \in \text{N}_{0}^{2}\text{c}ʻ\alpha .\equiv .\text{Cnv}ʻʻ\lambda \in ^{2}\text{N}_{0}\text{c}ʻ\alpha\)

*106·222. \(\vdash .\Lambda {\sim}\in ^{1}\text{N}_{0}\text{C}.^{1}\text{N}_{0}\text{C}\subset \text{N}_{0}\text{C}.^{1}\text{N}_{0}\text{C}\in \text{Cls ex}^{2}\,\text{excl} \quad[*106·22·212]\)

*106·223. \(\vdash .\Lambda {\sim}\in ^{2}\text{N}_{0}\text{C}.^{2}\text{N}_{0}\text{C}\subset \text{N}_{0}\text{C}.^{2}\text{N}_{0}\text{C}\in \text{Cls ex}^{2}\,\text{excl}\)

Other propositions of the same kind as the above may be proved by observing that, if \(m\) and \(n\) are indices for which \(t^{m}ʻ\alpha\) and \(t^{n}ʻ\alpha\) have been defined, we have \[ \gamma \subset t^{n}ʻ\alpha .\beta \in \text{N}^{m}\text{c}ʻ\alpha .\supset .\downarrow \beta ʻʻ\gamma \in \text{N}^{m}\text{c}ʻ\alpha , \] of which the proof is direct and simple. Hence, since we always have \(\exists !\text{N}^{m}\text{c}ʻ\alpha\), we also always have \[ \begin{aligned} &\exists !\text{N}^{m}\text{c}ʻ\alpha ,\\ \text{whence}\qquad \text{N}^{mm}\text{C}\subset \text{N}_{0}\text{C}.&\text{N}^{mm}\text{C}\in \text{Cls ex}^{2}\,\text{excl}. \end{aligned} \] We have in like manner \[ \exists !\text{N}_{0}^{m}\text{c}ʻ\alpha .\exists !^{m}\text{N}_{0}\text{c}ʻ\alpha. \] But we do not always have \[ \exists !\text{N}_{mn}\text{c}ʻ\alpha \,\,\text{or}\,\, \exists !\text{N}_{n}^{m}\text{c}ʻ\alpha \,\,\text{or}\,\, \exists !^{m}\text{N}_{n}\text{c}ʻ\alpha . \]

*106·23. \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} &\vdash .*64·33.*104·1.*63·5 .\supset \vdash :\text{Hp}.\supset .t^{11}ʻ\alpha =t_{00}ʻ\beta \\ &\vdash .(1).(*106·01·011).*100·321.\supset \vdash .\text{Prop} \end{array} \]

*106·231. \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta \quad[\text{Proof as in}\, *106·23]\)

*106·24. \(\vdash :\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta \quad[*106·23]\)

*106·241. \(\vdash :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta\)

The analogues of the above propositions for other indices or suffixes are similarly proved.

*106·25. \(\vdash .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\iotaʻʻ\alpha \quad[*106·23.*104·2]\)

*106·251. \(\vdash .\text{N}_{00}\text{c}ʻ\alpha =\text{N}_{11}\text{c}ʻ\iotaʻʻ\alpha\)

*106·31. \[\begin{aligned}\vdash :x,y\in t_{0}ʻ\alpha .t_{0}ʻ\alpha &=t_{0}ʻ\beta .x\neq y.\supset .\\ &\downarrow xʻʻ\alpha \in \text{N}_{00}\text{c}ʻ\alpha .\downarrow yʻʻ\beta \in \text{N}_{00}\text{c}ʻ\beta .\downarrow xʻʻ\alpha \cap \downarrow xʻʻ\beta =\Lambda \\ \quad[*106·2.*55·233]\end{aligned}\]

*106·311. \[\begin{aligned}\vdash \colon\ldotp x\in t_{0}ʻ\alpha .t_{0}ʻ\alpha &=t_{0}ʻ\beta :\alpha =\Lambda .\lor.\beta =\Lambda :\supset .\\ &\downarrow xʻʻ\alpha \in \text{N}_{00}\text{c}ʻ\alpha .\downarrow xʻʻ\beta \in \text{N}_{00}\text{c}ʻ\beta .\downarrow xʻʻ\alpha \cap \downarrow xʻʻ\beta =\Lambda \\ \quad[*106·2.*55·232.\text{Transp}]\end{aligned}\]

*106·312. \[\begin{aligned}\vdash :&t_{0}ʻ\alpha =\iota ʻx.\alpha =\beta =\iota ʻx.\supset .\\ &\iota ʻ(\iota ʻx\uparrow \iota ʻx)\in \text{N}_{00}\text{c}ʻ\alpha .\iota ʻ(\Lambda \uparrow \iota ʻx)\in \text{N}_{00}\text{c}ʻ\beta .\iota ʻ(\iota ʻx\uparrow \iota ʻx)\cap \iota ʻ(\Lambda \uparrow \iota ʻx)=\Lambda \end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*73·43.&\supset \vdash .\iota ʻ(\iota ʻx\uparrow \iota ʻx)\text{ sm }\iota ʻx.\iota ʻ(\Lambda \uparrow \iota ʻx)\text{ sm }\iota ʻx.\\ [*13·12] &\supset \vdash :\text{Hp}.\supset .\iota ʻ(\iota ʻx\uparrow \iota ʻx)\text{ sm }\alpha .\iota ʻ(\Lambda \uparrow \iota ʻx)\text{ sm }\beta &\qquad \text{(1)}\\ \vdash .*64·16.&\supset \vdash :\text{Hp}.\supset .\iota ʻx\uparrow \iota ʻx\in t_{00}ʻ\alpha .\Lambda \uparrow \iota ʻx\in t_{00}ʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*106·1.*51·161.*24·54.*55·202.&\supset \vdash .\text{Prop} \end{array} \]

[Pg 64]

*106·32. \(\vdash :t_{0}ʻ\alpha =t_{0}ʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*106·31.\supset \vdash \colon\ldotp \text{Hp}:(\exists x,y).x,y\in t_{0}ʻ\alpha .x\neq y:\supset .\\ (\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda &\qquad \text{(1)}\\ \vdash .*52·4. \supset \vdash :{\sim}(\exists x,y).x,y\in t_{0}ʻ\alpha .x\neq y.\supset .t_{0}ʻ\alpha \in 1\cup \iota ʻ\Lambda .\\ [*63·18] \supset .t_{0}\alpha \in 1 &\qquad \text{(2)}\\ \vdash .(2).*60·38.*63·105.*52·46.\supset \vdash :{\sim}(\exists x,y).x,y\in t_{0}ʻ\alpha .x\neq y.\exists !\alpha .\exists !\beta .\supset .\\ \alpha =\beta =t_{0}ʻ\alpha .t_{0}ʻ\alpha \in 1.\\ [*106·312] \supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda &\qquad \text{(3)}\\ \vdash .*106·311.*63·18.\supset \\ \vdash \colon\ldotp \text{Hp}:{\sim}(\exists !\alpha .\exists !\beta ):\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda &\qquad \text{(4)}\\ \vdash .(1).(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*106·4. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(00)}=\text{N}_{00}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*106·14.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \beta \in \mu _{00}.&\equiv :(\exists \gamma ).\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\beta \text{ sm }\gamma .\beta \in tʻt_{00}ʻ\gamma :\\ [*64·3] &\equiv :(\exists \gamma ).\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\beta \text{ sm }\gamma :\beta \in tʻt_{00}ʻ\alpha :\\ [*102·84] & \equiv :\beta \text{ sm }\alpha .\beta \in tʻt_{00}ʻ\alpha :\\ [*106·1] &\equiv :\beta \in \text{N}_{00}\text{c}ʻ\alpha \colon\colon \supset \vdash .\text{Prop} \end{array} \]

*106·401. \(\vdash :\mu =\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu _{(00)}=\text{N}^{11}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*104·24.*106·4.\supset \vdash :\text{Hp}.\supset .\mu _{(00)}&=\text{N}_{00}\text{c}ʻ\iotaʻʻ\alpha \\ [*106·25] &=\text{N}^{11}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*106·402. \(\vdash :\mu =\text{N}_{1}\text{c}ʻ\alpha .\exists !\mu .\supset .\mu _{(00)}=\text{N}_{11}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*106·231. \supset \vdash :\text{Hp}.\beta \in \mu .\supset .\text{N}_{11}\text{c}ʻ\alpha &=\text{N}_{00}\text{c}ʻ\beta \\ [*106·4.*103·26] & =\mu _{(00)} &\qquad \text{(1)}\\ \vdash .(1).*10·11·23·35.&\supset \vdash :\text{Hp}.\exists !\mu .\supset .\mu _{(00)}=\text{N}_{11}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*106·41. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(11)}=\text{N}^{11}\text{c}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*63·54.(*106·041).*103·27.\supset \\ \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \beta \in \mu ^{(11)}.&\equiv :(\exists \gamma ).\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\beta \text{ sm }\gamma :\beta \in tʻt^{11}ʻt_{0}ʻ\alpha :\\ [*102·84.*64·32] &\equiv :\beta \text{ sm }\alpha .\beta \in tʻt^{11}ʻ\alpha :\\ [(*106·011)] &\equiv :\beta \in \text{N}^{11}\text{c}ʻ\alpha \colon\colon \supset \vdash .\text{Prop} \end{array} \]

*106·411. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(11)}=\text{N}_{11}\text{c}ʻ\alpha \quad[\text{Proof as in}\, *106·41]\)

*106·43. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu _{(00)}.\delta \in \nu _{(00)}.\gamma \cap \delta =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*103·2.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\\ [*106·4] &\supset .(\exists \alpha ,\beta ).\mu _{(00)}=\text{N}_{00}\text{c}ʻ\alpha .\nu _{(00)}=\text{N}_{00}\text{c}ʻ\beta .\\ [*106·32] &\supset .(\exists \gamma ,\delta ).\gamma \in \mu _{(00)}.\delta \in \nu _{(00)}.\gamma \cap \delta =\Lambda :\supset \vdash .\text{Prop} \end{array} \]

[Pg 65]

*106·44. \(\vdash :\mu ,\nu \in \text{N}_{00}\text{C}.tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu .\delta \in \nu .\gamma \cap \delta =\Lambda \quad[*106·32]\)

The following propositions are analogous to *102·71 ff., and similar remarks apply to them.

*106·5. \[\begin{aligned}\vdash :R\in \text{Cls}\rightarrow &1.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR\subset \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ).\\ &W=\hat{x} \hat{y} \{x,y\in \alpha .{\sim}x(\breve{R} ʻx)y\}.\supset .W{\sim}\in \text{ᗡ}ʻR.W\unicode{x2abd} \alpha \uparrow \alpha\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*4·73.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp x,y\in \alpha .&\supset _{x,y}:xWy.\equiv .{\sim}x(\breve{R} ʻx)y:\\ [*5·18] &\supset _{x,y}:{\sim}\{xWy.\equiv .x(\breve{R} ʻx)y\}\colon\ldotp \\ [*10·1] \supset \colon\ldotp x\in \alpha .&\supset _{x}.{\sim}\{xWx.\equiv .x(\breve{R} ʻx)x\}.\\ [*21·43.\text{Transp}] &\supset _{x}.W\neq \breve{R} ʻx\colon\ldotp [\text{Hp}] \supset \colon\ldotp x\in \text{D}ʻR.\supset _{x}.W\neq \breve{R} ʻx\colon\ldotp \\ [*71·411.\text{Transp}] &\supset \colon\ldotp W{\sim}\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ \vdash .*21·33.(*35·04).&\supset \vdash :\text{Hp}.\supset .W\,\unicode{x2abd}\, \alpha \uparrow \alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*106·51. \(\vdash :\beta \subset \alpha .\supset .{\sim}\{\beta \text{ sm }\text{Rl}ʻ(\alpha \uparrow \alpha )\}\)

Dem. \[ \begin{array}{l} \vdash .*106·5. \supset \vdash :\text{Hp}.R\in 1\rightarrow 1.\text{D}ʻR=&\beta .\text{ᗡ}ʻR\subset \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ).\supset .\\ &(\exists W).W\in \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ).W{\sim}\in \text{ᗡ}ʻR.\\ [*13·14] &\supset .\text{ᗡ}ʻR\neq \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ) &\qquad \text{(1)}\\ \vdash .(1).*22·41.\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\in 1\rightarrow 1.\text{D}ʻR=\beta .\supset _{R}.\text{ᗡ}ʻR \neq \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ):\\ [*10·51.*73·1] &\supset :{\sim}{\beta \text{ sm }\text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha )}\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*106·52. \(\vdash :\beta \subset t_{0}ʻ\alpha .\supset .\beta {\sim}\in \text{Nc}ʻt_{00}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*106·51.\supset \vdash :\text{Hp}.&\supset .{\sim}\{\beta \text{ sm }\text{Rl}ʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha)\}.\\ [*64·54] &\supset .{\sim}\{\beta \text{ sm }t_{00}ʻ\alpha\}.\\ [*100·1] &\supset .\beta {\sim}\in \text{Nc}ʻt_{00}ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*106·53. \(\vdash .\text{Nc}(\alpha )ʻt_{00}ʻ\alpha =\Lambda \quad[*106·52.*102·6.*63·371]\)

*106·54. \(\vdash .\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha {\sim}\in \text{N}_{00}\text{C}\)

Dem. \[ \begin{array}{l} \vdash .*100·33.*103·15.&\supset \\ \vdash :\text{N}_{00}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .&\supset .\beta \text{ sm }t_{00}ʻ\alpha &\qquad \text{(1)}\\ \vdash .*103·12.(*106·01).&\supset \\ \vdash :\text{N}_{00}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .&\supset .t_{00}ʻ\alpha \in tʻt_{00}ʻ\beta .\\ [*63·16.(*64·01)] &\supset .tʻtʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha )=tʻtʻ(t_{0}ʻ\beta \uparrow t_{0}ʻ\beta ).\\ [*63·391] &\supset .tʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha )=tʻ(t_{0}ʻ\beta \uparrow t_{0}ʻ\beta ) .\\ [*64·3.(*64·01)] &\supset .t_{0}ʻ\alpha =t_{0}ʻ\beta .\\ [*63·105] &\supset .\beta \subset t_{0}ʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash :\text{N}_{00}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .\supset .\beta \in \text{Nc}ʻt_{00}ʻ\alpha .\beta \subset t_{0}ʻ\alpha &\qquad \text{(3)}\\ \vdash .(3).\text{Transp}.*106·52.&\supset \vdash .(\beta ).\text{N}_{00}\text{c}ʻ\beta \neq \text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .\\ [*106·13.\text{Transp}] &\supset \vdash .\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha {\sim}\in \text{N}_{00}\text{C}.\supset \vdash .\text{Prop} \end{array} \]

*106·55. \(\vdash .\exists !\text{N}_{0}\text{C}-\text{N}_{00}\text{C} \quad[*106·54]\)

Summary of Section B.

In the present section, we have to consider the arithmetical operations as applied to cardinals, as well as the relation of greater and less between cardinals. Thus the topics to be dealt with in this section are the first that can properly be said to belong to Arithmetic.

The treatment of addition, multiplication and exponentiation to be given in what follows is guided by the desire to secure the greatest possible generality. In the first place, everything to be said generally about the arithmetical operations must apply equally to finite and infinite classes or cardinals. In the second place, we desire such definitions as shall allow the number of summands in a sum or of factors in a product to be infinite. In the third place, we wish to be able to add or multiply two numbers which are not necessarily of the same type. In the fourth place, we wish our definitions to be such that the sum of the cardinal numbers of two or more classes shall depend only upon the cardinal numbers of those classes, and shall be the same when the classes overlap as when they are mutually exclusive; with similar conditions for the product. The desire to obtain definitions fulfilling all these conditions leads to somewhat more complicated definitions than would otherwise be required; but in the outcome, the result is simpler than if we started with simpler definitions, since we avoid vexatious exceptions.

The above observations will become clearer through their applications. Let us begin with the case of arithmetical addition of two classes.

If \(\alpha\) and \(\beta\) are mutually exclusive classes, the sum of their cardinal numbers will be the cardinal number of \(\alpha \cup \beta\). But in order that \(\alpha\) and \(\beta\) may be mutually exclusive, they must have no common members, and this is only significant when they are of the same type. Hence, given two perfectly general classes \(\alpha\) and \(\beta\), we require to find two classes which are mutually exclusive and are respectively similar to \(\alpha\) and \(\beta\); if these two classes are called \(\alpha'\) and \(\beta'\), then \(\text{Nc}ʻ(\alpha' \cup \beta')\) will be the sum of the cardinal numbers of \(\alpha\) and \(\beta\). We note that \(\Lambda \cap \alpha\) and \(\Lambda \cap \beta\) indicate respectively the \(\Lambda\)'s of the same types as \(\alpha\) and \(\beta\), and accordingly we take as \(\alpha'\) and \(\beta'\) the two classes \[ \downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha\,\,\text{and}\,\,(\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta ; \][Pg 67] these two classes are always of the same type, always mutually exclusive, and always similar to \(\alpha\) and \(\beta\) respectively. Hence we define \[ \alpha +\beta =\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \cup (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta \quad\text{Df}. \]

The sum of the cardinal numbers of \(\alpha\) and \(\beta\) will then be the cardinal number of \(\alpha + \beta\); hence we may call \(\alpha + \beta\) the arithmetical class-sum of two classes, in contradistinction to \(\alpha \cup \beta\), which is the logical sum. It will be noted that \(\alpha + \beta\), unlike \(\alpha \cup \beta\), does not require that \(\alpha\) and \(\beta\) should be of the same type. Also \(\alpha + \alpha\) is not identical with \(\alpha\), but when \(\alpha = \Lambda\), \(\alpha + \alpha\) is also \(\Lambda\), though in a different type. Thus the law of tautology does not hold of the arithmetical class-sum of two classes.

If \(\mu\) and \(\nu\) are two cardinals of assigned types, we denote their arithmetical sum by \(\mu +_{\text{c}} \nu\). (As many kinds of arithmetical addition occur in our work, and as it is essential to our purpose to distinguish them, we effect the distinction by suffixes to the sign of addition. It is, of course, only in dealing with principles that these different symbols are needed: we do not wish to suggest that they should be adopted in ordinary mathematics.) Now if \(\mu +_{\text{c}} \nu\) is to have the properties which we commonly associate with the sum of two cardinals, it must be typically ambiguous, and must be the cardinal number of any class which can be divided into two mutually exclusive parts having \(\mu\) terms and \(\nu\) terms respectively. Hence we are led to the following definition: \[ \mu +_{\text{c}}\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}. \]

In this definition, various points should be noted. In the first place, it does not require that \(\mu\) and \(\nu\) should be of the same type; \(\mu +_{\text{c}} \nu\) is significant whenever \(\mu\) and \(\nu\) are classes of classes. Thus it is not necessary for significance that \(\mu\) and \(\nu\) should be cardinals, though if they are not both cardinals, \(\mu +_{\text{c}} \nu = \Lambda\). If they are both cardinals, we find \[ \mu +_{\text{c}}\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha +\beta )\}. \]

Thus in this case \(\alpha \in \mu .\beta \in \nu .\supset .\alpha+\beta \in \mu +_{\text{c}}\nu\).

Hence if neither \(\mu\) nor \(\nu\) is null, and if \(\alpha\) has \(\mu\) terms and \(\beta\) has \(\nu\) terms, \(\alpha + \beta\) is a member of \(\mu +_{\text{c}} \nu\). It easily follows that \[ \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\supset .\mu +_{\text{c}}\nu =\text{Nc}ʻ(\alpha +\beta ). \]

Hence when \(\mu\) and \(\nu\) are homogeneous cardinals (i.e. when they are cardinals other than \(\Lambda\)), their sum is the number of the arithmetical class-sum of any two classes having \(\mu\) terms and \(\nu\) terms respectively.

A few words are necessary to explain why, in the definition, we put \(\mu = \text{N}_{0}\text{c}ʻ\alpha . \nu =\text{N}_{0}\text{c}ʻ\beta\) rather than \(\mu = \text{Nc}ʻ\alpha .\nu = \text{Nc}ʻ\beta\). The reason is this. Suppose either \(\mu\) or \(\nu\), say \(\mu\), is \(\Lambda\). Then, by *102·73, \(\mu = \text{Nc}(\zeta)ʻtʻ\zeta\), if \(\zeta\) is of the appropriate type. Hence if we had put \[ \mu +_{\text{c}}\nu =\hat{\xi} \{(\exists \alpha , \beta ).\mu =\text{Nc}ʻ\alpha .\nu =\text{Nc}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}, \][Pg 68] where the ambiguities of type involved in \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) may be determined as we please, we should have \[ \begin{aligned} \nu &=\text{Nc}ʻ\beta .\supset .tʻ\zeta +\beta \in \mu +_{\text{c}}\nu ,\\ \textit{i.e.}\qquad \nu &=\text{Nc}ʻ\beta .\supset .tʻ\zeta +\beta \in \Lambda +_{\text{c}} \nu . \end{aligned} \] We should also have \(tʻtʻ\zeta +\beta \in \Lambda +_{\text{c}}\nu\) and so on. Thus \(\Lambda +_{\text{c}}\nu\) would not have a definite value, i.e. it would not merely have typical ambiguity, which it ought to have, but it would not have a definite value even when its type was assigned. Thus such a definition would be unsuitable. For the above reasons, we put \(\mu =\text{N}_{0}\text{c}ʻ\alpha . \nu =\text{N}_{0}\text{c}ʻ\beta\) in the definition, and obtain the typical ambiguity which we desire by means of the typical ambiguity of the "\(\text{sm}\)" in "\(\xi \text{ sm }(\alpha +\beta)\)." It is always essential to right symbolism that the values of typically ambiguous symbols should be unique as soon as their type is assigned. The scope of these definitions and of the corresponding definitions for multiplication and exponentiation (*113·04·05.*116·03·04) is extended by convention \(\text{IIT}\) of the prefatory statement.

The above definition of \(\mu +_{\text{c}}\nu\) is designed for the case in which \(\mu\) and \(\nu\) are typically definite. But we must be able to speak of "\(\text{Nc}ʻ\gamma+_{\text{c}}\text{Nc}ʻ\delta\)," and this must be a definite cardinal, namely \(\text{Nc}ʻ(\gamma +\delta)\). If we simply write \(\text{Nc}ʻ\gamma , \text{Nc}ʻ\delta\) in place of \(\mu\), \(\nu\) in the definition of \(\mu +_{\text{c}}\nu\), we find \[ \text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta =\hat{\xi} \{(\exists \alpha ,\beta ).\text{Nc}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha .\text{Nc}ʻ\delta =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\}. \] But this will not always have a definite value when the type of \(\text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta\) is assigned. To take a simple case, write \(tʻ\zeta\) for \(\gamma\) and \(tʻy\) for \(\delta\). Then \[ \text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻtʻy=\hat{\xi} \{(\exists \alpha ,\beta ).\text{Nc}ʻtʻ\zeta =\text{N}_{0}\text{c}ʻ\alpha .\text{Nc}ʻtʻy=\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\}. \] whence we easily obtain \[ \text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻtʻy=\hat{\xi} \{(\exists \alpha ).\text{Nc}ʻtʻ\zeta =\text{N}_{0}\text{c}ʻ\alpha .\xi \text{ sm }(\alpha +\iota ʻy)\}. \] If we determine the ambiguity of \(\text{Nc}ʻtʻ\zeta\) to be \(\text{N}_{1}\text{c}ʻtʻ\zeta\), we find \[ \text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻ\iota ʻy=\Lambda \] in all types; but if we determine the ambiguity to be \(\text{N}_{0}\text{c}ʻtʻ\zeta\), we have \[ \text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻ\iota ʻy=\text{Nc}ʻ(tʻ\zeta +\iota ʻy), \] and this exists in the type of \(tʻ\zeta +\iota ʻy\), if not in lower types. Hence the value of \(\text{Nc}ʻtʻ\zeta+_{\text{c}}\text{Nc}ʻ\iota ʻy\) depends upon the determination of the ambiguity of \(\text{Nc}ʻtʻ\zeta\). It is obvious that we want our definition to yield \[ \text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta =\text{Nc}ʻ(\gamma +\delta ) \] in all types; but in order to insure that this shall hold even when, for some values of \(\zeta\), \(\text{Nc}(\zeta )ʻ\gamma =\Lambda\), we must introduce two new definitions, namely \[ \begin{aligned} \text{Nc}ʻ\alpha +_{\text{c}}\mu &=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\mu \quad\text{Df},\\ \mu +_{\text{c}}\text{Nc}ʻ\alpha &=\mu +_{\text{c}}\text{N}_{0}\text{c}ʻ\alpha \quad\text{Df},\\ \text{whence}\qquad \vdash :\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta &=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta =\text{Nc}ʻ(\alpha +\beta ). \end{aligned} \] This definition is to be applied when "\(\text{Nc}ʻ\gamma\)" and "\(\text{Nc}ʻ\delta\)" occur without any[Pg 69] determination of type. On the other hand, if we have \(\text{Nc}(\zeta)ʻ\gamma\) and \(\text{Nc}(\eta)ʻ\delta\), we apply the definition of \(\mu +_{\text{c}}\nu\). We shall find that whenever \(\text{Nc}(\zeta )ʻ\gamma\) and \(\text{Nc}(\eta )ʻ\delta\) both exist, \[ \text{Nc}(\zeta )ʻ\gamma +_{\text{c}} \text{Nc}(\eta )ʻ\delta =\text{N}_{0}\text{c}ʻ\gamma +_{\text{c}} \text{N}_{0}\text{c}ʻ\delta . \] Thus the above definition is only required in order to exclude values of \(\zeta\) or \(\eta\) for which either \(\text{Nc}(\zeta )ʻ\gamma\) or \(\text{Nc}(\eta )ʻ\delta\) is \(\Lambda\).

The commutative and associative laws of arithmetical addition are easily deduced from the definition of \(\alpha +\beta\). We shall have \[ \begin{aligned} \vdash .\alpha &+\beta =\text{Cnv}ʻʻ(\beta +a),\\ \text{whence}\qquad \vdash .\text{Nc}ʻ\alpha &+_{\text{c}}\text{Nc}ʻ\beta =\text{Nc}ʻ\beta +_{\text{c}}\text{Nc}ʻ\alpha , \end{aligned} \] because each = \(\text{Nc}ʻ(\alpha +\beta)\). A similar though slightly longer proof shows that \[ \begin{aligned} \vdash .(\alpha +\beta )&+\gamma \text{ sm }\alpha +(\beta +\gamma ),\\ \text{whence}\qquad \vdash .(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta )&+_{\text{c}}\text{Nc}ʻ\gamma =\text{Nc}ʻ\alpha +_{\text{c}}(\text{Nc}ʻ\beta +_{\text{c}}\text{Nc}ʻ\gamma). \end{aligned} \]

The above definition of \(\alpha +\beta\) enables us to proceed to the sum of any finite number of classes, and allows any one class to recur in the summation. But it does not enable us to define the sum of an infinite number of classes. For this we need a new definition. Since an infinite number of classes cannot be given by enumeration, but only by intension, we shall have to take a class of classes \(\kappa\), and define the arithmetical sum of the members of \(\kappa\). Thus now the classes which are the summands must all be of the same type (since they are all members of \(\kappa\)), and no one class can occur more than once, since each member of \(\kappa\) only counts once. (In order to deal with repetition, we must advance to multiplication, which will be explained shortly.) Thus in removing the limitation to a finite number of summands, we introduce certain other limitations. This is the reason which makes it worth while to introduce the above definition of \(\alpha +\beta\) in addition to the definition now to be given.

If \(\kappa\) is a class of classes, the sum of the cardinal numbers of the members of \(\kappa\) will evidently be obtained by constructing a class of mutually exclusive classes whose members have a one-one relation to the members of corresponding members of \(\kappa\). Suppose \(\alpha\), \(\beta\) are two different members of \(\kappa\), and suppose \(x\) is a member both of \(\alpha\) and of \(\beta\). Then we wish to count \(x\) twice over, once as a member of \(\alpha\) and once as a member of \(\beta\). The simplest way to do this is to form the ordinal couples \(x\downarrow \alpha\) and \(x\downarrow \beta\), which are not identical except when \(\alpha\) and \(\beta\) are identical. Thus if we take all such ordinal couples, i.e. if we take the class \[ \hat{R} \{(\exists x).x\in \alpha .R=x\downarrow \alpha\}, \] for every \(\alpha\) which is a member of \(\kappa\), we get a class of mutually exclusive classes, namely the classes of the form \(\downarrow\alpha ʻʻ\alpha\), where \(\alpha \in \kappa\), each of these is similar to the corresponding member of \(\kappa\). Hence the logical sum of this class of classes, i.e. \[ \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .x\in \alpha .R=x\downarrow \alpha\}, \][Pg 70] has the required number of terms. Now, by *85·601, \[ \downarrow \alpha ʻʻ\alpha =\in \unicode{x21A7}ʻ\alpha . \] Hence the class whose logical sum we are taking is \(\in\unicode{x21A7}ʻʻ\kappa\). Hence we put \[ \Sigma ʻ\kappa =sʻ\in \unicode{x21A7}ʻʻ\kappa \quad\text{Df}. \] \(\Sigma ʻ\kappa\) may be called the arithmetical sum of \(\kappa\), in contradistinction to \(sʻ\kappa\), which is the logical sum. Thus \(\Sigma ʻ\kappa\) bears to \(sʻ\kappa\) a relation analogous to that which \(\alpha +\beta\) bears to \(\alpha \cup \beta\).

We put further \(\Sigma \text{Nc}ʻ\kappa =\text{Nc}ʻsʻ\in\unicode{x21A7}ʻʻ\kappa \quad\text{Df}.\)

Thus \(\Sigma \text{Nc}ʻ\kappa\) is the sum of the numbers of members of \(\kappa\).

It is to be observed that \(\Sigma \text{Nc}ʻ\kappa\) is not in general a function of \(\text{Nc}ʻʻ\kappa\). For, if two members of \(\kappa\) have the same cardinal number, this will only count once in \(\text{Nc}ʻʻ\kappa\), whereas it counts twice in \(\Sigma \text{Nc}ʻ\kappa\).

We shall find that, provided \(\alpha \neq \beta\), \[ \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )=\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta . \] Thus where a finite number of summands are concerned, the two definitions of addition agree, except that the first allows one class to count several times over, while the second does not.

In dealing with multiplication, our procedure is closely analogous to the procedure for addition. We first define the arithmetical class-product of two classes \(\alpha\) and \(\beta\), which is a certain class whose cardinal number is the product of the cardinal numbers of \(\alpha\) and \(\beta\). We write \(\beta \times \alpha\) for the arithmetical class-product of \(\beta\) and \(\alpha\), and define it as the class of all ordinal couples of which the referent is a member of \(\alpha\) and the relatum a member of \(\beta\), i.e. as \[ \hat{R} \{(\exists x,y).x\in \alpha .y\in \beta .R=x\downarrow y\}. \] By *40·7, this class is \(sʻ\alpha \downarrow_{,,}ʻʻ\beta\). Hence we put \[ \beta \times \alpha =sʻ\alpha \downarrow_{,,}ʻʻ\beta \quad\text{Df}. \] The class \(\alpha \downarrow_{,,}ʻʻ\beta\) is similar to \(\beta\), and each member of it is similar to \(\alpha\); hence if \(\text{N}_{0}\text{c}ʻ\alpha =\mu\) and \(\text{N}_{0}\text{c}ʻ\beta=\nu\), \(sʻ\alpha \downarrow_{,,}ʻʻ\beta\) consists of \(\nu\) classes having \(\mu\) members each. The class \(\alpha\downarrow_{,,}ʻʻ\beta\) is important also in connection with exponentiation.

The product of two cardinals is defined as follows: \[ \mu \times _{\text{c}}\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha \times \beta )\} \quad\text{Df}. \] In regard to types, this definition calls for analogous remarks to those which were made on \(\mu +_{\text{c}}\nu\). Also, as before, we need definitions of \(\mu \times _{\text{c}}\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\alpha \times _{\text{c}}\mu\), whence we obtain \[ \text{Nc}ʻ\alpha \times _{\text{c}}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha \times _{\text{c}}\text{N}_{0}\text{c}ʻ\beta \quad\text{Df}. \] [Pg 71]By means of these definitions, we can define the product of any finite number of cardinals; but in order to define products which have an infinite number of factors, we need a new definition.

If \(\kappa\) is a class of classes, we take \({\in}_{\Delta}ʻ\kappa\) as its arithmetical product. In simple cases, it is easy to see the justification of this decision. E.g. let \(\kappa\) consist of the three classes \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), and let the members of \(\alpha _{1}\) be \(x_{1}\), \(x_{2}\); those of \(\alpha _{2}\), \(y_{1}\), \(y_{2}\); those of \(\alpha _{3}\), \(z_{1}\), \(z_{2}\). Then the members of \({\in}_{\Delta}ʻ\kappa\) are \[ \begin{aligned} &x_{1}\downarrow \alpha _{1}\unicode{x228d} y_{1}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ &x_{2}\downarrow \alpha _{1}\unicode{x228d} y_{1}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ &x_{1}\downarrow \alpha _{1}\unicode{x228d} y_{2}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ &x_{2}\downarrow \alpha _{1}\unicode{x228d} y_{2}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ \end{aligned} \] with four more obtained by substituting \(z_{2}\) for \(z_{1}\) in the above. Thus \(\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa = 8 = \text{Nc}ʻ\alpha_{1} \times _{\text{c}} \text{Nc}ʻ\alpha _{2} \times _{\text{c}}\text{Nc}ʻ\alpha _{3}\). In general, however, the existence of \({\in}_{\Delta}ʻ\kappa\) is doubtful, owing to the doubt as to the validity of the multiplicative axiom. (We shall return to this point shortly.) Hence there is no proof that the product of an infinite number of factors cannot be zero unless one of the factors is zero.

When \(\kappa\) is a class of mutually exclusive classes, \({\in}_{\Delta}ʻ\kappa\) is similar to \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\). On account of its lower type, \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is often more convenient than \({\in}_{\Delta}ʻ\kappa\). Hence we put \[ \text{Prod}ʻ\kappa =\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}, \] or (what comes to the same thing) \[ \text{Prod}=\text{D}_{\in }\mid {\in}_{\Delta} \quad\text{Df}. \] For the product of the cardinal numbers of the members of \(\kappa\), we put \[ \Pi \text{Nc}ʻ\kappa =\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}. \] As in the case of \(\Sigma \text{Nc}ʻ\kappa\), \(\Pi \text{Nc}ʻ\kappa\) is not in general a function of \(\text{Nc}ʻʻ\kappa\). We shall have \[ \vdash :\alpha \neq \beta .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )=\text{Nc}ʻ\alpha \times _{\text{c}}\text{Nc}ʻ\beta . \] Thus for products of a finite number of different factors, the two definitions of multiplication agree.

It remains to define exponentiation. Since this is not a commutative operation, it essentially involves an order as between the base and the exponent; hence we do not obtain a definition of the exponentiation of a class \(\kappa\), analogous to \(\Sigma \text{Nc}ʻ\kappa\) or \(\Pi\text{Nc}ʻ\kappa\), but only a definition of \(\mu ^\nu\), which may be extended to any finite number of exponentiations. We put \[ \alpha\,\text{exp}\,\beta =\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \quad\text{Df}. \] where \(\alpha \downarrow_{,,}ʻʻ\beta\) has the meaning explained above, resulting from *38·03. It will be observed that, if \(\text{N}_{0}\text{c}ʻ\alpha = \mu\) and \(\text{N}_{0}\text{c}ʻ\beta= \nu\), \(\alpha \downarrow_{,,}ʻʻ\beta\) is a class of \(\nu\) mutually exclusive classes each of which has \(\mu\) terms; hence \(\alpha \text{exp}\beta\) may suitably be used to define \(\mu ^\nu\). Hence we put \[ \mu ^\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha \,\text{exp}\,\,\beta )\} \quad\text{Df}, \] and for the same reasons as before, we put \[ \text{Nc}ʻ\alpha ^{\text{N}_{c}\text{c}ʻ\beta} = \text{N}_{0}\text{c}ʻ\alpha ^{\text{N}_{0}\text{c}ʻ\beta} \quad\text{Df}. \]

[Pg 72]

The above definition of exponentiation gives the same value of \(\mu^\nu\) as results from Cantor's definition by means of "Belegungen." The class of Cantor's "Belegungen" is \[ \begin{aligned} \hat{R} \{R\in 1\rightarrow &\text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR=\beta \},\\ \textit{i.e.}\qquad &(\alpha \uparrow \beta )_{\Delta }ʻ\beta, \end{aligned} \] and it is easily proved that this is similar to \(\alpha \,\text{exp}\,\beta\).

The usual formal properties of exponentiation result without much difficulty from the above definitions.

The above definition of exponentiation is so framed as to make propositions on exponentiation independent of the multiplicative axiom, except when exponentiation is to be connected with multiplication, i.e. when it is to be shown that the product of \(\nu\) factors, each of which is \(\mu\), is \(\mu ^\nu\). This proposition cannot be proved generally without the multiplicative axiom. Similarly, in the theory of multiplication, the proposition that the sum of \(\nu \mu\)'s is \(\mu \times _{\text{c}} \nu\) requires the multiplicative axiom (as does also the proposition that a product is zero when and only when one of its factors is zero). Otherwise, the theory of multiplication proceeds without the need for employing the multiplicative axiom.

To take first the connection of addition and multiplication: this connection, in the form in which we naturally suppose it to hold, is affirmed in the proposition: \[ \begin{aligned} \mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cls excl}ʻ\mu .&\supset .sʻ\kappa \in \mu \times _{\text{c}}\nu &\qquad \text{(A)}\\ \textit{or}\qquad \mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .&\supset .\Sigma ʻ\kappa \in \mu \times _{\text{c}}\nu . \end{aligned} \] We will take the first of these as being simpler. It affirms that the sum of \(\nu \mu\)'s is \(\mu \times _{\text{c}}\nu\). This can be proved when \(\nu\) is finite, whether \(\mu\) is finite or not; but when \(\nu\) is infinite, it cannot be proved without the multiplicative axiom. This may be seen as follows. We know that \[ \begin{aligned} \vdash :&\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\supset .\\ &\alpha \downarrow_{,,}ʻʻ\beta \in \nu \cap \text{Cls excl}ʻ\mu .sʻ\alpha \downarrow_{,,}ʻʻ\beta \in \mu \times _{\text{c}}\nu &\qquad \text{(A)}\\ \end{aligned} \] Thus (A) above will result if we can prove \[ \kappa ,\lambda \in \nu \cap \text{Cls excl}ʻ\mu .\supset .sʻ\kappa \text{ sm }sʻ\lambda , \] since we shall put \(\alpha \downarrow_{,,}ʻʻ\beta\) for \(\lambda\), and use (B).

Since \(\kappa\), \(\lambda \in \nu\), we have \(\kappa\text{ sm }\lambda\). Assume \[ S\in 1\rightarrow 1.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda . \] Let \(\kappa _{1}\), \(\kappa _{2}\), ... be members of \(\kappa\), and let \(\lambda _{1}\), \(\lambda _{2}\), ... be the members of \(\lambda\) which are correlated with \(\kappa _{1}\), \(\kappa_{2}\), ... by \(S\), i.e. \(\lambda _{1}=\breve{S}ʻ\kappa _{1} . \lambda _{2}=\breve{S} ʻ\kappa _{2}.\text{etc}\). We have, since \(\kappa\), \(\lambda \in \text{Cl}ʻ\mu\), [Pg 73]\(\kappa _{1}\text{ sm }\lambda _{1}.\kappa _{2}\text{ sm }\lambda_{2}.\text{etc}\). Thus \(\alpha S\beta .\supset _{\alpha ,\beta}.\alpha \text{ sm }\beta\), i.e. \(S\,\unicode{x2abd}\, \text{ sm }\). If \(\kappa\) and \(\lambda\) are finite, we can pick out arbitrarily a correlation \(S_{1}\) for \(\kappa _{1}\) and \(\lambda _{1}\), another \(S_{2}\) for \(\kappa _{2}\) and \(\lambda _{2}\), and so on; then \(S_{1}\unicode{x228d} S_{2}\unicode{x228d}\) ... correlates \(sʻ\kappa\) and \(sʻ\lambda\), and therefore \(sʻ\kappa \text{ sm }sʻ\lambda\). But when \(\kappa\) and \(\lambda\) are infinite, this method is impracticable. In this case, we proceed as follows.

By *73·01, \(\alpha \overline{\text{ sm }} \beta =(1\rightarrow 1)\cap\overleftarrow{\text{D}}ʻ\alpha \cap \overleftarrow{\text{ᗡ}}ʻ\beta\quad\text{Df}\).

Thus "\(\alpha \overline{\text{ sm }} \alpha\)" will stand for all the permutations of a class into itself; "\(\alpha \overline{\text{ sm }}\beta\)" stands for all the permutations of \(\alpha\) into \(\beta\), i.e. all the \(1\rightarrow 1\)'s whose domain is \(\alpha\) and whose converse domain is \(\beta\). It is obvious that \[ \vdash :\exists !\alpha \overline{\text{ sm }} \beta \cap \gamma \overline{\text{ sm }} \delta .\supset .\alpha =\gamma .\beta =\delta . \] In the case of the \(\kappa\) and \(\lambda\) above, we know that \(\alpha \text{ sm }\beta\) when \(\alpha S\beta\); thus \[ \begin{aligned} \alpha \in \kappa .\supset _{\alpha }.\exists !\alpha \overline{\text{ sm }} (\breve{S} ʻ\alpha )\\ \text{or}\qquad\qquad\quad \beta \in \lambda .\supset _{\beta }.\exists !(Sʻ\beta )\overline{\text{ sm }} \beta .\\ \text{Put}\qquad \text{Crp}(S)ʻ\beta =(Sʻ\beta )\overline{\text{ sm }} \beta \qquad\text{Df}, \end{aligned} \] where "\(\text{Crp}\)" stands for "correspondence." Thus \(\text{Crp}(S)ʻ\beta\) is the class of all correspondences of \(Sʻ\beta\) and \(\beta\); \(\text{Crp}(S)ʻʻ\lambda\) is the class of all such classes of correspondences. If we extract one member out of each of these classes of correspondences, we get a class of relations whose sum is a correlator of \(sʻ\kappa\) and \(sʻ\lambda\); i.e. \[ \varpi \in \text{D}ʻʻ{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\supset .\dot{s} ʻ\varpi \in (sʻ\kappa )\overline{\text{ sm }} (sʻ\lambda ). \] Thus the desired result follows whenever \[ \exists !{\in}_{\Delta }ʻ\text{Crp}(S)ʻʻ\lambda . \] Now we have \(S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\supset .\text{Crp}(S)ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\).

Consequently \[ \text{Mult ax}.\supset :S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda .\kappa ,\lambda \in \text{Cls}^{2} \text{excl}. \supset .sʻ\kappa \text{ sm }sʻ\lambda , \] whence, by what was said previously, \[ \text{Mult ax}.\supset :\kappa \in \nu \cap \text{Cls excl}ʻ\mu .\supset .sʻ\kappa \in \mu \times _{\text{c}}\nu .\Sigma \text{Nc}ʻ\kappa =\mu \times _{\text{c}}\nu . \]

The consideration of \({\in}_{\Delta }ʻ\text{Crp}(S)ʻʻ\lambda\) leads similarly to the proposition \[ \vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .{\in}_{\Delta}ʻ\kappa \in \mu ^\nu .\Pi \text{Nc}ʻ\kappa =\mu ^\nu. \] The proof is closely analogous to that for the connection of addition and multiplication.

It will be seen that, in the above use of the multiplicative axiom, we have two classes of classes \(\kappa\) and \(\lambda\) concerning which we assume \[ (\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda , \] i.e. we assume that \(\kappa\) and \(\lambda\) are similar classes of similar classes. A slightly modified hypothesis concerning \(\kappa\) and \(\lambda\) will enable us to obtain many results, without the multiplicative axiom, which otherwise might be expected to require this axiom. This is effected as follows.

[Pg 74]

Put \(\kappa \text{ sm } \text{ sm } \lambda .\equiv .(\exists T).T\in 1\rightarrow 1.\text{ᗡ}ʻT=sʻ\lambda .\kappa =T_{\in }ʻʻ\lambda\), where "\(\text{ sm }\text{ sm }\)" is a single symbol representing a relation.

When this relation holds between \(\kappa\) and \(\lambda\), we shall say that \(\kappa\) and \(\lambda\) have "double similarity." In this case, \(T\) correlates \(sʻ\kappa\) and \(sʻ\lambda\), while \(T_{\in}\) correlates \(\kappa\) and \(\lambda\), so that if \(\beta\) is a member of \(\lambda\), \(T_{\in }ʻ\beta\), i.e. \(Tʻʻ\beta\), is its correlate in \(\kappa\). We shall then have \[ \begin{aligned} &\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .sʻ\kappa \text{ sm } sʻ\lambda ,\\ &\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .\Sigma \text{Nc}ʻ\kappa =\Sigma \text{Nc}ʻ\lambda ,\\ &\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .\Pi \text{Nc}ʻ\kappa =\Pi \text{Nc}ʻ\lambda . \end{aligned} \]

Also we have \[ \vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda . \] Conversely, \[ \begin{aligned} &\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda .\\ &\varpi \in \text{D}ʻʻ{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .T=\dot{s} ʻ\varpi .\supset .T\in 1\rightarrow 1.\text{ᗡ}ʻT=sʻ\lambda .\kappa =T_{\in }ʻʻ\lambda , \end{aligned} \] whence \[ \begin{aligned} \vdash \colon\colon \text{Mult ax}.\supset \colon\ldotp \kappa ,\lambda \in \text{Cls}^{2} \text{excl}:&(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda :\\ &\supset .\kappa \text{ sm } \text{ sm } \lambda . \end{aligned} \] Hence the multiplicative axiom is only required in order to pass from \[ (\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda \] to \(\kappa \text{ sm } \text{ sm } \lambda\). It is this fact, and the consequent possibility of diminishing the use of the multiplicative axiom, which has led us to the employment of "\(\text{sm} \,\text{sm}\)" in the present section.

We treat also, in this section, the relation of greater and less between cardinals. We say that \(\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta\) when there is a part of \(\alpha\) which is similar to \(\beta\), but no part of \(\beta\) is similar to \(\alpha\). The principal proposition in this subject is the Schröder-Bernstein theorem, i.e. \[ \vdash :\mu \geq \nu .\nu \geq \mu .\supset .\mu =\nu . \] This is an immediate consequence of *73·88. It cannot be shown, without assuming the multiplicative axiom, that of any two cardinals one must be the greater, i.e. \[ \mu ,\nu \in \text{NC}.\mu \neq \nu .\supset :\mu >\nu .\lor.\nu >\mu. \] If we assume the multiplicative axiom, this results from Zermelo's proof that on that assumption, every class can be well-ordered, together with Cantor's proof that of any two well-ordered series which are not similar, one must be similar to a part of the other. But these propositions cannot be proved till a much later stage (*258).

Summary of *110.

In this number, we start from the definition:

*110·01. \(\alpha +\beta =\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \cup (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta \quad\text{Df}\)

\(\alpha +\beta\) is called the "arithmetical class-sum" of \(\alpha\) and \(\beta\). The definition is framed so as to give two mutually exclusive classes respectively similar to \(\alpha\) and \(\beta\), so that the number of terms in the logical sum of these two classes is the arithmetical sum of the numbers of terms in \(\alpha\) and \(\beta\) respectively. \(\alpha +\beta\) is significant whenever \(\alpha\) and \(\beta\) are classes, whatever their types may be.

By means of \(\alpha +\beta\), we define the arithmetical sum of two cardinals as follows:

*110·02. \(\mu +_{\text{c}} \nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}\)

This defines the "arithmetical sum of two cardinals." (It is not necessary to significance that \(\mu\) and \(\nu\) should be cardinals, but only that they should be classes of classes. If, however, either is not a cardinal, \(\mu +_{\text{c}}\nu =\Lambda\)). It will be observed that, when \(\mu\) and \(\nu\) are typically definite, so are \(\alpha\) and \(\beta\) in the above definition; but \(\xi\) is typically ambiguous, on account of the ambiguity of "\(\text{ sm }\)." Hence \(\mu +_{\text{c}}\nu\) is also typically ambiguous.

It will be shown that \(\mu +_{\text{c}}\nu\) is always a cardinal, and that, if \[ \mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta, \quad\text{then}\quad \mu +_{\text{c}}\nu =\text{Nc}ʻ(\alpha +\beta ). \] Hence whenever \(\mu\) and \(\nu\) are cardinals other than \(\Lambda\), \(\mu +_{\text{c}}\nu\) is an existent cardinal in some types, though it may be \(\Lambda\) in others.

Two more definitions are required in this number, namely:

*110·03. \(\text{Nc}ʻ\alpha +_{\text{c}}\mu =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\mu \quad\text{Df}\)

*110·04. \(\mu +_{\text{c}} \text{Nc}ʻ\alpha =\mu +_{\text{c}}\text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)

These definitions are needed in order to apply the definition of \(\mu +_{\text{c}}\nu\) to the case in which \(\mu\) and \(\nu\) are replaced by typically ambiguous symbols \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\). It does not make any difference to the value of \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta\) how the ambiguities of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are determined, so long as they are determined in a way that insures \(\exists !\text{Nc}ʻ\alpha.\exists !\text{Nc}ʻ\beta\); but if there are types in which either \(\text{Nc}ʻ\alpha\) or \(\text{Nc}ʻ\beta\) is \(\Lambda\), we get \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta =\Lambda\) in all types if we determine the ambiguities so that \(\text{Nc}ʻ\alpha=\Lambda\) or \(\text{Nc}ʻ\beta =\Lambda\). It is in order to[Pg 76] exclude such determinations of the ambiguity that the above definitions are required. Also in connection with these definitions and the corresponding definitions *113·04·05 and *116·03·04 and *117·02·03, the convention \(\text{IIT}\) of the prefatory statement must be noted.

The propositions of the present number begin with the properties of \(\alpha + \beta\). We show (*110·11·12) that \(\alpha + \beta\) consists of two mutually exclusive parts, which are respectively similar to \(\alpha\) and \(\beta\); we show (*110·14) that if \(\alpha\) and \(\beta\) are mutually exclusive, \(\alpha \cup \beta\) is similar to \(\alpha + \beta\), and (*110·15) that if \(\gamma\) and \(\delta\) are respectively similar to \(\alpha\) and \(\beta\), then \(\gamma + \delta\) is similar to \(\alpha + \beta\). We show (*110·16) that \(\text{Nc}ʻ(\alpha + \beta)\) consists of all classes which can be divided into two mutually exclusive parts which are respectively similar to \(\alpha\) and \(\beta\).

We then proceed (*110·2—·252) to the consideration of \(\mu +_{\text{c}} \nu\). Here \(\mu\) and \(\nu\) are typically definite, and the definition *110·02 applies to any typically definite symbols, such as \(\text{N}_{0}\text{c}ʻ\alpha\) or \(\text{Nc}(\eta )ʻ\alpha\). We prove (*110·21) that if \(\mu\) and \(\nu\) are cardinals, their sum consists of all classes similar to some class of the form \(\alpha + \beta\), where \(\alpha \in \mu .\beta \in \nu\); we prove (*110·22) that the sum of \(\text{N}_{0}\text{c}ʻ\alpha\) and \(\text{N}_{0}\text{c}ʻ\beta\) is \(\text{Nc}ʻ(\alpha + \beta)\), and (*110·25) that if \(\mu\) and \(\nu\) are cardinals, their sum is equal to the sum of the "same" cardinals in any other types in which they are not null, i.e.

*110·25. \(\vdash :\mu ,\nu \in \text{NC}.\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu +_{\text{c}}\nu = \text{ sm }_{\eta }ʻʻ\mu +_{\text{c}}\text{ sm }_{\zeta }ʻʻ\nu \)

We then (*110·3—·351) consider \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta\), to which we apply the definitions *110·03·04. We have

*110·3. \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta = \text{Nc}ʻ(\alpha + \beta)\) whence the other properties of \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta\) follow from previous propositions.

We then have (*110·4—·44) various propositions on the type of \(\mu +_{\text{c}}\nu\) and its existence and kindred matters. The chief of these are

*110·4. \(\vdash :\exists !\mu +_{\text{c}}\nu .\supset .\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C}\)

*110·42. \(\vdash .\mu +_{\text{c}}\nu \in \text{NC}\)

This proposition requires no hypothesis, because, if \(\mu\) and \(\nu\) are not both cardinals, \(\mu +_{\text{c}}\nu = \Lambda\), and \(\Lambda\) is a cardinal, by *102·74.

Our next set of propositions (*110·5—·57) are concerned with the permutative and associative laws, which are *110·51 and *110·56 respectively.

We then (*110·6—·643) consider the addition of 0 or 1, proving (*110·61) that a cardinal is unchanged by the addition of 0, and (*110·643) that \(1 +_{\text{c}}1 = 2\).

*110·01. \(\alpha +\beta = \downarrow (\Lambda \cap \beta )ʻʻ\iota ʻʻ\alpha \cup (\Lambda \cap \alpha ) \downarrow ʻʻ\iota ʻʻ\beta \quad\text{Df}\)

*110·02. \(\mu +_{\text{c}}\nu = \hat{\xi} \{(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}\)

*110·03. \(\text{Nc}ʻ\alpha +_{\text{c}}\mu = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\mu \quad\text{Df}\)

*110·04. \(\mu +_{\text{c}}\text{Nc}ʻ\alpha = \mu +_{\text{c}}\text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)

[Pg 77]

These definitions are extended by \(\text{IIT}\) of the prefatory statement.

*110·1. \[\begin{aligned}\vdash \colon\ldotp R\in \alpha +\beta .\equiv :&(\exists x).x\in \alpha .R = (\iota ʻx)\downarrow (\Lambda \cap \beta ).\lor.\\ &(\exists y).y\in \beta .R = (\Lambda \cap \alpha )\downarrow (\iota ʻy)\\ &[*38·13·131 .(*110·01)]\end{aligned}\]

*110·101. \(\vdash .(\iota ʻx)\downarrow (\Lambda \cap \beta ) \neq (\Lambda \cap \alpha )\downarrow (\iota ʻy)\)

Dem. \[ \begin{array}{l} \vdash .*55·15. &\supset \vdash .\text{D}ʻ(\iota ʻx)\downarrow (\Lambda \cap \beta ) = \iota ʻ\iota ʻx.\text{D}ʻ(\Lambda \cap \alpha )\downarrow (\iota ʻy) = \iota ʻ(\Lambda \cap \alpha ) &\qquad \text{(1)}\\ \vdash . *51·161. &\supset \vdash .\iota ʻx \neq (\Lambda \cap \alpha ).\\ [*51·23] &\supset \vdash .\iota ʻ\iota ʻx \neq \iota ʻ(\Lambda \cap \alpha ) &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash .\text{D}ʻ(\iota ʻx)\downarrow (\Lambda \cap \beta ) \neq \text{D}ʻ(\Lambda \cap \alpha )\downarrow (\iota ʻy).\supset \vdash .\text{Prop} \end{array} \]

*110·11. \(\vdash .\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \cap (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*110·101.&\supset \vdash :x\in \alpha .R = \downarrow (\Lambda \cap \beta )ʻ\iota ʻx.y\in \beta .S = (\Lambda \cap \alpha )\downarrow ʻ\iota ʻy.\supset .R \neq S:\\ [*37·67] &\supset \vdash :R\in \downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha .S\in (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta .\supset .R \neq S &\qquad \text{(1)}\\ \vdash .(1).*24·37.\supset \vdash .\text{Prop} \end{array} \]

*110·12. \(\vdash .\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \text{ sm }\alpha .(\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta \text{ sm }\beta \quad[*73·41·61·611]\)

*110·11·12 give the justification for the use of \(\alpha + \beta\) in defining arithmetical addition, since they show that \(\alpha + \beta\) consists of two mutually exclusive parts which are respectively similar to \(\alpha\) and \(\beta\).

*110·13. \(\vdash :\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\supset .\gamma \cup \delta \text{ sm }(\alpha +\beta)\)

Dem. \[ \begin{array}{l} \vdash .*110·12.\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha .\delta \text{ sm }(\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*110·11.*73·71.\supset \vdash .\text{Prop} \end{array} \]

*110·14. \(\vdash :\alpha \cap \beta = \Lambda .\supset .\alpha \cup \beta \text{ sm }(\alpha +\beta ) \quad[*110·13.*73·3]\)

Thus whenever \(\alpha\) and \(\beta\) are mutually exclusive, their logical sum may replace their arithmetical sum in defining the sum of their cardinal numbers.

*110·15. \(\vdash :\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\supset .\gamma +\delta \text{ sm }\alpha +\beta\)

Dem. \[ \begin{array}{l} \vdash .*110·12.\supset \vdash :\text{Hp}.\supset .\downarrow (\Lambda \cap \delta )ʻʻ\iotaʻʻ\gamma \text{ sm }\alpha .(\Lambda \cap \gamma )\downarrow ʻʻ\iotaʻʻ\text{ sm }\beta &\qquad \text{(1)}\\ \vdash .*110·11.\supset \vdash .\downarrow (\Lambda \cap \delta )ʻʻ\iotaʻʻ\gamma \cap (\Lambda \cap \gamma )\downarrow ʻʻ\iotaʻʻ\delta = \Lambda &\qquad \text{(2)}\\ \vdash .(1).(2).*110·13.\supset \\ \vdash :\text{Hp}.\supset .\downarrow (\Lambda \cap \delta )ʻʻ\iotaʻʻ\gamma \cup (\Lambda \cap \gamma )\downarrow ʻʻ\iotaʻʻ\delta \text{ sm }\alpha +\beta :\supset \vdash .\text{Prop} \end{array} \]

*110·151. \(\vdash \colon\ldotp \alpha \cap \beta = \Lambda .\supset :\xi \text{ sm }(\alpha \cup \beta ).\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta\)

Dem. \[ \begin{array}{l} \vdash .*73·71.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ &(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta .\supset .\xi \text{ sm }(\alpha \cup \beta ) &\qquad \text{(1)}\\ \vdash .*72·411.*37·25·22.*73·22.\supset \\ \vdash :S\in &1\rightarrow 1.\text{D}ʻS = \xi .\text{ᗡ}ʻS = \alpha \cup \beta .\alpha \cap \beta = \Lambda .\supset .\\ &Sʻʻ\alpha \cap Sʻʻ\beta = \Lambda .\xi = Sʻʻ\alpha \cup Sʻʻ\beta .Sʻʻ\alpha \text{ sm }\alpha .Sʻʻ\beta \text{ sm }\beta .\\ [*11·36] &\supset .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta &\qquad \text{(2)}\\ \vdash .(2).*10·11·23·35.*73·1.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\xi \text{ sm }(\alpha \cup \beta ).\supset .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta &\qquad \text{(3)}\\ \vdash .(1).(3).\supset \vdash .\text{Prop} \end{array} \]

[Pg 78]

*110·152. \(\vdash :\xi \text{ sm } (\alpha +\beta ). \equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta\)

Dem. \[ \begin{array}{l} \vdash .*110·151·11.\supset \\ \vdash :\xi \text{ sm } (\alpha +\beta ). &\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \downarrow (\Lambda \cap \beta )ʻʻ\iota ʻʻ\alpha .\delta \text{ sm } (\Lambda \cap \alpha )\downarrow ʻʻ\iota ʻʻ\beta .\\ &\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta .\\ [*73·37.*110·12] &\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta :\supset \vdash .\text{Prop} \end{array} \]

*110·16. \[\begin{aligned}&\vdash .\text{Nc}ʻ(\alpha +\beta ) = \hat{\xi} \{(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta \}\\ &[*110·152.*100·1]\end{aligned}\]

*110·17. \(\vdash :\alpha \in tʻ\beta .\supset .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha +\beta )\)

Dem. \[ \begin{array}{l} \vdash . *104·43.\supset \\ \vdash :\text{Hp}.&\supset .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\gamma \subset tʻ\alpha .\delta \text{ sm } \beta .\delta \subset tʻ\alpha .\gamma \cap \delta = \Lambda .\\ [*22·59] & \supset .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\gamma \cup \delta \subset tʻ\alpha .\\ [*110·16] &\supset .(\exists \xi ).\xi \subset tʻ\alpha .\xi \in \text{Nc}ʻ(\alpha +\beta ).\\ [*102·6.*63·5] &\supset .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha +\beta ):\supset \vdash .\text{Prop} \end{array} \]

Thus when \(\alpha\) and \(\beta\) are of the same type, \(\text{Nc}ʻ(\alpha +\beta )\) exists at least in the type next above that of \(\alpha\) and \(\beta\). We cannot prove that it exists in the type of \(\alpha\) and \(\beta\). E.g. suppose the lowest type contained only one member; then if \(x\) were that one member, \(\text{Nc}ʻ(\iota ʻx+\iota ʻx)\) would not exist in the type to which \(\iota ʻx\) belongs, but would exist in the next type, i.e. there would not be two individuals, but there would be two classes, namely \(\Lambda\) and \(\iota ʻx\), so that \(\iota ʻ\Lambda \cup \iota ʻ\iota ʻx\in \text{Nc}ʻ(\iota ʻx+\iota ʻx)\).

*110·18. \(\vdash .\alpha +\beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta)\)

Dem. \[ \begin{array}{l} \vdash .*64·53. &\supset \vdash :x\in \alpha .\supset .\downarrow (\Lambda \cap \beta )ʻ\iota ʻx\in tʻ(tʻ\alpha \uparrow tʻ\beta ) &\qquad \text{(1)}\\ \vdash .(1).*37·61.&\supset \vdash .\downarrow (\Lambda \cap \beta )ʻʻ\iota ʻʻ\alpha \subset tʻ(tʻ\alpha \uparrow tʻ\beta ) &\qquad \text{(2)}\\ \text{Similarly}\quad &\vdash .(\Lambda \cap \alpha )\downarrow ʻʻ\iota ʻʻ\beta \subset tʻ(tʻ\alpha \uparrow tʻ\beta ) &\qquad \text{(3)}\\ \vdash .(2).(3). &\supset \vdash .\alpha +\beta \subset tʻ(tʻ\alpha \uparrow tʻ\beta ).\\ [*63·5] & \supset \vdash .\alpha +\beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta ).\supset \vdash .\text{Prop} \end{array} \]

*110·2. \[\begin{aligned}&\vdash :\xi \in \mu +_{\text{c}}\nu . \equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\alpha .\xi \text{ sm } (\alpha +\beta )\\ &[(*110·02)]\end{aligned}\]

*110·201. \[\begin{aligned}&\vdash \colon\ldotp \xi \in \mu +_{\text{c}}\nu . \equiv :\mu ,\nu \in \text{NC}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha +\beta )\\ &[*103·27.*110·2]\end{aligned}\]

*110·202. \[\begin{aligned}\vdash \colon\ldotp \xi \in &\mu +_{\text{c}}\nu . \equiv :\\ &\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*110·2·152 .&\supset \vdash \colon\ldotp \xi \in \mu +_{\text{c}}\nu . \equiv :\\ (\exists \alpha ,\beta ,\gamma ,\delta ).&\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta :\\ [*103·28] &\equiv :(\exists \gamma ,\delta ).\exists !\mu .\exists !\nu .\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta \colon\ldotp \\ &\supset \vdash .\text{Prop} \end{array} \]

[Pg 79]

*110·21. \[\begin{aligned}&\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu +_{\text{c}}\nu .\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha +\beta )\\ &[*110·201]\end{aligned}\]

*110·211. \[\begin{aligned}\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi &\in \mu +_{\text{c}}\nu .\equiv .\\ &(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ\nu .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*110·21·152.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu +_{\text{c}}\nu .\equiv .\\ &(\exists \alpha ,\beta ,\gamma ,\delta ).\alpha \in \mu .\beta \in \nu .\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ [*37·1] &\equiv .(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ\nu .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*110·212. \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu +_{\text{c}}\nu .\equiv .(\exists \gamma ).\gamma \in \text{ sm }ʻʻ\mu .\gamma \subset \xi .\xi -\gamma \in \text{ sm }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash .*110·211.*24·47.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu +_{\text{c}}\nu .&\equiv .(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ\nu .\gamma \subset \xi .\delta =\xi -\gamma .\\ [*13·195] &\equiv .(\exists \gamma ).\gamma \in \text{ sm }ʻʻ\mu .\gamma \subset \xi .\xi -\gamma \in \text{ sm }ʻʻ\nu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*110·22. \(\vdash .\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta = \text{Nc}ʻ(\alpha + \beta )\)

Dem. \[ \begin{array}{l} \vdash .*103·4.*110·211.\supset \\ \vdash :\xi \in \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta .&\equiv .(\exists \gamma ,\delta ).\gamma \in \text{Nc}ʻ\alpha .\delta \in \text{Nc}ʻ\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ [*100·31] &\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ [*110·16] &\equiv .\xi \in \text{Nc}ʻ(\alpha + \beta ):\supset \vdash .\text{Prop} \end{array} \]

*110·221. \(\vdash :\xi \in \text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta .\equiv .\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\xi \in \text{Nc}ʻ(\alpha +\beta )\)

Dem. \[ \begin{array}{l} \vdash .*110·202.&\supset \vdash \colon\ldotp \xi \in \text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta .\\ &\equiv :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta :(\exists \gamma ,\delta ).\text{Nc}(\eta )ʻ\alpha =\text{Nc}ʻ\gamma .\\ &\text{Nc}(\zeta )ʻ\beta =\text{Nc}ʻ\delta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta :\\ [*100·35]&\equiv :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta :(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta :\\ [*110·16]&\equiv :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\xi \in \text{Nc}ʻ(\alpha +\beta )\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*110·23. \[\begin{aligned}\vdash :&\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\supset .\\ &\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta =\text{Nc}ʻ(\alpha +\beta )=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta \quad[*110·221·22]\end{aligned}\]

Thus \(\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta\) is independent of \(\eta\) and \(\zeta\) so long as \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) exist in the types of \(\eta\) and \(\zeta\) respectively.

*110·231. \[\begin{aligned}&\vdash \colon\ldotp \text{Nc}(\eta )ʻ\alpha =\Lambda .\lor.\text{Nc}(\zeta )ʻ\beta =\Lambda :\supset .\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta =\Lambda\\ &[*110·221]\end{aligned}\]

*110·24. \(\vdash :\eta \text{ sm }\alpha .\zeta \text{ sm }\beta .\supset .\text{N}_{0}\text{c}ʻ\eta +_{\text{c}}\text{N}_{0}\text{c}ʻ\zeta =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*103·42. \supset \vdash :\text{Hp}.&\supset.\text{N}_{0}\text{c}ʻ\eta =\text{Nc}(\eta )ʻ\alpha .\text{N}_{0}\text{c}ʻ\zeta =\text{Nc}(\zeta )ʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*103·13.\supset \vdash :\text{Hp}.&\supset .\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\\ [*110·23] &\supset .\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 80]

*110·25. \(\vdash :\mu ,\nu \in \text{NC}.\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu +_{\text{c}}\nu = \text{ sm }_{\eta }ʻʻ\mu +_{\text{c}} \text{ sm }_{\zeta }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash .*103·27.\supset \vdash :\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\\ \supset .\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\\ [*103·41.*102·85]\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta . \text{ sm }_{\eta }ʻʻ\mu = \text{Nc}(\eta )ʻ\alpha .\\ \text{ sm }_{\zeta }ʻʻ\nu = \text{Nc}(\zeta )ʻ\beta .\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\\ [*110·23] \supset .\mu +_{\text{c}}\nu = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta = \text{ sm }_{\eta }ʻʻ\mu +_{\text{c}} \text{ sm }_{\zeta }ʻʻ\nu &\qquad \text{(1)}\\ \vdash .(1).*10·11·23·35.\supset \\ \vdash :\mu ,\nu \in \text{NC}.\exists !\mu .\exists !\nu .\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu +_{\text{c}}\nu = \text{ sm }_{\eta }ʻʻ\mu +_{\text{c}} \text{ sm }_{\zeta }ʻʻ\nu &\qquad \text{(2)}\\ \vdash .*37·29.\text{Transp}.\supset \vdash :\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\supset .\exists !\mu .\exists !\nu &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*110·251. \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \mu +_{\text{c}}\nu\)

Dem. \[ \begin{array}{l} \vdash *110·25.*104·265.\supset \\ \vdash :\text{Hp}.\exists !\mu ^{(1)}.\exists !\nu ^{(1)}.&\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \mu +_{\text{c}}\nu &\qquad \text{(1)}\\ \vdash .*110·202.&\supset \vdash :{\sim}(\exists !\mu ^{(1)}.\exists !\nu ^{(1)}).\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \Lambda &\qquad \text{(2)}\\ \vdash .*104·264.\supset \vdash :\text{Hp}(2).&\supset .{\sim}(\exists !\mu .\exists !\nu ).\\ [*110·202] &\supset .\mu +_{\text{c}}\nu = \Lambda .\\ [(2)] &\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \mu +_{\text{c}}\nu &\qquad \text{(3)}\\ \vdash .(1).(3).\supset \vdash .\text{Prop} \end{array} \]

*110·252. \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu _{(00)}+_{\text{c}}\nu _{(00)} = \mu +_{\text{c}}\nu \quad[\text{Proof as in *110·251}]\)

A similar proof applies to \(\mu ^{(2)}\), \(\nu ^{(2)}\), etc., and to any such derived cardinals whose existence follows from that of \(\mu\) and \(\nu\). The proposition does not hold generally for \(\mu _{(1)}\), \(\nu _{(1)}\) and other descending derived cardinals, because they may be null when \(\mu\) and \(\nu\) exist.

The following proposition (*110·3) is more often used than any other in this number except *110·4.

*110·3. \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta = \text{Nc}ʻ(\alpha +\beta ) \quad[*110·22.(*110·03·04)]\)

*110·31. \(\vdash :\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\supset .\text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta = \text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta \quad[*110·24·3]\)

The following proposition is frequently used.

*110·32. \(\vdash :\alpha \cap \beta = \Lambda .\supset .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta = \text{Nc}ʻ(\alpha \cup \beta ) \quad[*110·3·14]\)

*110·33. \[\begin{aligned}&\vdash :\xi \in \text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta . \equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta \\ &[*110·3·16]\end{aligned}\]

The above proposition is used in *110·63. We might have used the above to define arithmetical addition, but this method would have been less convenient than the method adopted in this number, both because there would[Pg 81] have been more difficulty in dealing with types, and because the existence of \(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta\) (in the types in which it does exist) is less evident with the above definition than with the definitions given in *110·01·02·03·04.

*110·331. \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta =\hat{\xi} \{(\exists \gamma ).\gamma \text{ sm }\alpha .\xi -\gamma \text{ sm }\beta .\gamma \subset \xi\}\)

Dem. \[ \begin{array}{l} \vdash . *110·33.*24·47.&\supset \\ \vdash :\xi \in \text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta .&\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \subset \xi .\delta =\xi -\gamma .\\ [*13·195] &\equiv .(\exists \gamma ).\gamma \text{ sm }\alpha .\xi -\gamma \text{ sm }\beta .\gamma \subset \xi :\supset \vdash .\text{Prop} \end{array} \]

*110·34. \[\begin{aligned}&\vdash :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\supset .\text{Nc}(\eta )ʻ\alpha +_{\text{c}} \text{Nc}(\zeta )ʻ\beta =\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta \\ &[*110·23·3]\end{aligned}\]

*110·35. \(\vdash .\text{N}^{1}\text{c}ʻ\alpha +_{\text{c}} \text{N}^{1}\text{c}ʻ\beta =\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta \quad[*104·102·21.*110·34]\)

*110·351. \(\vdash .\text{N}_{00}\text{c}ʻ\alpha +_{\text{c}} \text{N}_{00}\text{c}ʻ\beta =\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta \quad[*106·21.*110·34]\)

Similar propositions will hold generally for ascending cardinals.

The following proposition (*110·4) is the most used of the propositions in this number. It is useful both in the form given, and in the form resulting from transposition, in which it shows that \(\mu +_{\text{c}}\nu =\Lambda\) unless both \(\mu\) and \(\nu\) are existent cardinals. It is chiefly useful in avoiding the necessity of the hypothesis \(\mu\), \(\nu \in \text{NC}\) in such propositions as the commutative and associative laws.

*110·4. \(\vdash :\exists !\mu +_{\text{c}} \nu .\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C} \quad[*110·201·202·2]\)

The following propositions, down to *110·411 inclusive, are concerned with types. They are not referred to in the sequel.

*110·401. \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\supset .\alpha + \beta \in tʻtʻ(\mu \uparrow \nu )\)

Dem. \[ \begin{array}{l} \vdash .*110·18.*103·12.\supset \vdash :\text{Hp}.&\supset .\alpha + \beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta ).\alpha \in \mu .\beta \in \nu .\\ [*63·11] &\supset .\alpha + \beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta ).tʻ\alpha =t_{0}ʻ\mu .tʻ\beta =t_{0}ʻ\nu .\\ [*13·12] &\supset .\alpha + \beta \in tʻtʻ(t_{0}ʻ\mu \uparrow t_{0}ʻ\nu ).\\ [*64·13] &\supset .\alpha + \beta \in tʻtʻ(\mu \uparrow \nu ):\supset \vdash .\text{Prop} \end{array} \]

*110·402. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\exists !(\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu )\)

Dem. \[ \begin{array}{l} \vdash .*110·22.*100·3.\supset \\ \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .&\supset .\alpha + \beta \in \mu +_{\text{c}} \nu .\\ [*110·401] &\supset .\alpha + \beta \in (\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu ).\\ [*10·24] &\supset .\exists !(\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu ) (1) \vdash .(1).*103·2.\supset \vdash .\text{Prop} \end{array} \]

*110·403. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\equiv .\exists !(\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu ) \quad[*110·402·4]\)

[Pg 82]

*110·404. \(\vdash .\exists !(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta )\cap tʻtʻ(tʻ\alpha \uparrow tʻ\beta ) \quad[*110·18·3.*100·3]\)

*110·41. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.tʻ\mu = tʻ\nu .\supset .\exists !(\mu +_{\text{c}}\nu )\cap tʻ\mu \)

Dem. \[ \begin{array}{l} \vdash .*103·11.\supset \vdash :&\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .tʻ\mu = tʻ\nu .\supset .\\ &\mu \subset tʻ\alpha .\nu \subset tʻ\beta .tʻ\mu = tʻ\nu .\\ [*63·21·35] &\supset .t_{0}ʻ\mu = tʻ\alpha .t_{0}ʻ\nu = tʻ\beta .t_{0}ʻ\mu = t_{0}ʻ\nu .\\ [*13·16·17] &\supset .tʻ\alpha = tʻ\beta = t_{0}ʻ\mu .\\ [*110·17] &\supset .\exists !\text{Nc}ʻ(\alpha +\beta )\cap tʻt_{0}ʻ\mu .\\ [*110·22.*63·19] &\supset .\exists !(\mu +_{\text{c}}\nu )\cap tʻ\mu :\supset .\text{Prop} \end{array} \]

*110·411. \[\begin{aligned}&\vdash :tʻ\alpha = tʻ\beta .\supset .\exists !(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta )\cap tʻtʻ\alpha .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha +\beta )\\ &[*110·17·3]\end{aligned}\]

It will be observed that the following proposition (*110·42) requires no hypothesis. This is owing to *110·4 and *102·74.

*110·42. \(\vdash .\mu +_{\text{c}}\nu \in \text{NC}\)

Dem. \[ \begin{array}{l} \vdash .*110·22. \supset \vdash :\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .&\supset .\mu +_{\text{c}}\nu = \text{Nc}ʻ(\alpha +\beta ).\\ [*100·41] &\supset .\mu +_{\text{c}}\nu \in \text{NC}&\qquad \text{(1)}\\ \vdash .(1).*103·2. &\supset \vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\mu +_{\text{c}}\nu \in \text{NC}&\qquad \text{(2)}\\ \vdash .*110·4.\text{Transp}.\supset .\vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).&\supset .\mu +_{\text{c}}\nu = \Lambda .\\ [*102·74] &\supset .\mu +_{\text{c}}\nu \in \text{NC} &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*110·43. \(\vdash :\mu +_{\text{c}}\nu = \text{N}_{0}\text{c}ʻ\eta . \equiv .\eta \in \mu +_{\text{c}}\nu \quad[*110·42.*103·26]\)

*110·44. \(\vdash .\text{ sm }ʻʻ(\mu +_{\text{c}}\nu ) = \mu +_{\text{c}}\nu\)

Dem. \[ \begin{array}{l} \vdash .*37·1.*110·2.\supset \\ \vdash :\xi \in \text{ sm }ʻʻ(\mu +_{\text{c}}\nu ). &\equiv .(\exists \eta ,\alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\eta \text{ sm }(\alpha +\beta ).\xi \text{ sm }\eta .\\ [*73·3·32] &\equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta ).\\ [*110·2] &\equiv .\xi \in \mu +_{\text{c}}\nu :\supset \vdash .\text{Prop} \end{array} \]

The above proposition depends upon the fact that \(\mu +_{\text{c}}\nu\) is typically ambiguous, even when \(\mu\) and \(\nu\) are typically definite. It is used in the theory of inductive cardinals (*120·32·41·424).

The following propositions are concerned with the commutative and associative laws for arithmetical addition of cardinals.

*110·5. \(\vdash .\beta +\alpha = \text{Cnv}ʻʻ(\alpha +\beta)\)

Dem. \[ \begin{array}{l} \vdash .*55·14.\supset \vdash .\text{Cnv}ʻʻ(\alpha +\beta ) &= \Lambda _{\beta }\downarrow ʻʻ\iota ʻʻ\alpha \cup \downarrow \Lambda _{\alpha }ʻʻ\iota ʻʻ\beta \\ [(*110·01)] &= \beta +\alpha .\supset \vdash .\text{Prop} \end{array} \]

[Pg 83]

*110·501. \(\vdash .\beta + \alpha \text{ sm }\alpha + \beta \quad[*110·5.*73·4]\)

*110·51. \(\vdash .\mu +_{\text{c}} \nu =\nu +_{\text{c}} \mu \quad[*110·2·501.*73·37]\)

It is not necessary to the truth of the above proposition that \(\mu\) and \(\nu\) should be cardinals. If either is not a cardinal, \(\mu +_{\text{c}} \nu\) and \(\nu +_{\text{c}} \mu\) are both \(\Lambda\).

The following propositions lead to the associative law (*110·56).

*110·52. \[\begin{aligned}\vdash :\xi \text{ sm }(\alpha + \beta ) + \gamma .&\equiv .(\exists \pi ,\rho ,\sigma ).\pi \text{ sm }\alpha .\rho \text{ sm }\beta .\sigma \text{ sm }\gamma .\\ &\pi \cap \rho =\Lambda .\pi \cap \sigma =\Lambda .\rho \cap \sigma =\Lambda .\xi =\pi \cup \rho \cup \sigma\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*110·152.\supset \vdash \colon\ldotp \xi \text{ sm }(\alpha + \beta ) + \gamma .\equiv :(\exists \eta ,\sigma ).&\eta \text{ sm }(\alpha + \beta ).\\ &\sigma \text{ sm }\gamma .\eta \cap \sigma =\Lambda .\xi =\eta \cup \sigma :\\ [*110·152] \equiv :(\exists \pi ,\rho ,\eta ,\sigma ).\pi \text{ sm }\alpha .&\rho \text{ sm }\beta .\pi \cap \rho =\Lambda .\eta =\pi \cup \rho .\\ &\sigma \text{ sm }\gamma .\eta \cap \sigma =\Lambda .\xi =\eta \cup \sigma :\\ [*13·195.*22·68.*24·32] &\equiv :(\exists \pi ,\rho ,\sigma ).\pi \text{ sm }\alpha .\rho \text{ sm }\beta .\sigma \text{ sm }\gamma .\pi \cap \rho =\Lambda .\\ &\pi \cap \sigma =\Lambda .\rho \cap \sigma =\Lambda .\xi =\pi \cup \rho \cup \sigma \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*110·521. \[\begin{aligned}\vdash :&\xi \text{ sm }\alpha + (\beta + \gamma ).\equiv .(\exists \pi ,\rho ,\sigma ).\pi \text{ sm }\alpha .\rho \text{ sm }\beta .\sigma \text{ sm }\gamma .\\ &\pi \cap \rho =\Lambda .\pi \cap \sigma =\Lambda .\rho \cap \sigma =\Lambda .\xi =\pi \cup \rho \cup \sigma \quad[*110·501·52]\end{aligned}\]

*110·53. \(\vdash .(\alpha + \beta )+ \gamma \text{ sm }\alpha + (\beta + \gamma ) \quad[*110·52·521]\)

*110·531. \(\alpha + \beta + \gamma =(\alpha + \beta ) + \gamma \quad\text{Df}\)

*110·54. \(\vdash .(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta ) +_{\text{c}} \text{Nc}ʻ\gamma =\text{Nc}ʻ(\alpha + \beta + \gamma )\)

Dem. \[ \begin{array}{l} \vdash .*110.3.\supset \vdash .(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta ) +_{\text{c}} \text{Nc}ʻ\gamma &=\text{Nc}ʻ(\alpha + \beta ) +_{\text{c}} \text{Nc}ʻ\gamma \\ [*110·3.(*110·531)] & =\text{Nc}ʻ(\alpha + \beta + \gamma ).\supset \vdash .\text{Prop} \end{array} \]

*110·541. \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}} (\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ\gamma )=\text{Nc}ʻ(\alpha + \beta + \gamma )\)

Dem. \[ \begin{array}{l} \vdash .*110·3.\supset \vdash .\text{Nc}ʻ\alpha +_{\text{c}} (\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ\gamma )&=\text{Nc}ʻ{\alpha + (\beta + \gamma )}\\ [*110·53.(*110·531)] & =\text{Nc}ʻ(\alpha + \beta + \gamma ).\supset \vdash .\text{Prop} \end{array} \]

*110·55. \(\vdash .(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta ) +_{\text{c}} \text{Nc}ʻ\gamma =\text{Nc}ʻ\alpha +_{\text{c}} (\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ\gamma ) \quad[*110·54·541]\)

*110·551. \[\begin{aligned}&\vdash .(\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}} \text{N}_{0}\text{c}ʻ\beta ) +_{\text{c}} \text{N}_{0}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}} (\text{N}_{0}\text{c}ʻ\beta +_{\text{c}} \text{N}_{0}\text{c}ʻ\gamma )\\ &[*110·55.(*110·03·04)]\end{aligned}\]

*110·56. \(\vdash .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\mu +_{\text{c}} (\nu +_{\text{c}} \varpi )\)

Dem. \[ \begin{array}{l} \vdash .*110·551.*103·2.\supset \\ \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.&\supset .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\mu +_{\text{c}} (\nu +_{\text{c}} \varpi ) &\qquad \text{(1)}\\ \vdash .*110·4.\text{Transp}.&\supset \\ \vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}).&\supset .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\Lambda .\mu +_{\text{c}} (\nu +_{\text{c}} \varpi )=\Lambda .\\ [*13.171] &\supset .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\mu +_{\text{c}} (\nu +_{\text{c}} \varpi )&\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 84]

This is the associative law for arithmetical addition. It will be seen that, like the commutative law, it does not require that \(\mu\), \(\nu\), \(\varpi\) should be cardinals.

*110·561. \(\mu +_{\text{c}} \nu +_{\text{c}} \varpi = (\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi \quad\text{Df}\)

*110·57. \(\vdash .(\mu +_{\text{c}} \nu ) +_{\text{c}} (\varpi +_{\text{c}} \rho ) = \mu +_{\text{c}} \nu +_{\text{c}} \varpi +_{\text{c}} \rho \quad[*110·56.(*110·561)]\)

The following propositions, concerning the addition of 0 or 1, are used frequently in dealing with inductive cardinals (*120).

*110·6. \(\vdash :\mu \in \text{NC}.\supset .\mu +_{\text{c}} 0 = \text{ sm }ʻʻ\mu\)

Dem. \[ \begin{array}{l} \vdash .*101·11.*110·21.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu +_{\text{c}} 0.&\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in 0.\xi \text{ sm }(\alpha + \beta ).\\ [*54·102] &\equiv .(\exists \alpha ).\alpha \in \mu .\xi \text{ sm }(\alpha + \Lambda ).\\ [*110·152] & \equiv .(\exists \alpha ,\gamma ,\delta ).\alpha \in \mu .\gamma \text{ sm }\alpha .\delta \text{ sm }\Lambda .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ [*73·47] &\equiv .(\exists \alpha ,\gamma ).\alpha \in \mu .\gamma \text{ sm }\alpha .\xi =\gamma .\\ [*13·195] &\equiv .(\exists \alpha ).\alpha \in \mu .\xi \text{ sm }\alpha .\\ [*37·1] &\equiv .\xi \in \text{ sm }ʻʻ\mu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

When \(\mu\) is a typically definite cardinal, \(\text{ sm }ʻʻ\mu\) is the same cardinal rendered typically ambiguous; when \(\mu\) is a typically ambiguous cardinal, \(\text{ sm }ʻʻ\mu\), is \(\mu\). In place of the above proposition, we might write \(\mu \in \text{NC}.\supset.\mu +_{\text{c}} 0 = \mu\); this would be true whenever the ambiguity of \(\mu +_{\text{c}} 0\) was so determined as to make it significant. But the above form gives more information.

*110·61. \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}} 0=\text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*101·1.\supset \vdash .\text{Nc}ʻ\alpha +_{\text{c}} 0&=\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\Lambda \\ [*110·32] &= \text{Nc}ʻ(\alpha \cup \Lambda )\\ [*24·24] &= \text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} \end{array} \]

In this proposition, \(\text{Nc}ʻ\alpha\) is typically ambiguous; hence we escape the necessity of putting \(\text{ sm }ʻʻ\text{Nc}ʻ\alpha\) on the right, as we should have to do if \(\text{Nc}ʻ\alpha\) were typically definite. We can deduce *110·61 from *110·6 as follows: \[ \begin{array}{l} \vdash .*110·3.\supset \vdash .\text{Nc}ʻ\alpha +_{\text{c}} 0&=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}} 0\\ [*110·6] &=\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \\ [*103·4] &=\text{Nc}ʻ\alpha \end{array} \]

[Pg 85]

We have to travel via \(\text{N}_{0}\text{c}ʻ\alpha\) in this proof, in order to avoid the possibility of a typical determination of \(\text{Nc}ʻ\alpha\) which would make \(\text{Nc}ʻ\alpha=\Lambda\). It is for the same reason that we cannot put "\(\text{ sm }ʻʻ\text{Nc}ʻ\alpha =\text{Nc}ʻ\alpha\)"; for if the first \(\text{Nc}ʻ\alpha\) is determined to a type in which \(\text{Nc}ʻ\alpha =\Lambda\), while the second is not, this equation becomes false.

*110·62. \(\vdash :\mu +_{\text{c}} \nu =0.\equiv .\mu =0.\nu =0\)

Dem. \[ \begin{array}{l} \vdash .*103·27.*101·11·13.&\supset \vdash .0=\text{N}_{0}\text{c}ʻ\Lambda &\quad \text{(1)}\\ \vdash .(1).*110·43.\supset \\ \vdash \colon\ldotp \mu +_{\text{c}} \nu =0.&\equiv :\Lambda \in \mu +_{\text{c}} \nu :\\ [*110·202] &\equiv :\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu =\text{Nc}ʻ\gamma .\nu =\text{Nc}ʻ\delta .\gamma \cap \delta =\Lambda .\gamma \cup \delta =\Lambda :\\ [*24·32.*13·22] &\equiv :\exists !\mu .\exists !\nu :\mu =\text{Nc}ʻ\Lambda .\nu =\text{Nc}ʻ\Lambda :\\ [*101·1·12] &\equiv :\mu =0.\nu =0\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*110·63. \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}} 1=\hat{\xi} \{(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}\)

Dem. \[ \begin{array}{l} \vdash .*101·2.\supset \\ \vdash .\text{Nc}ʻ\alpha +_{\text{c}} 1&=\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\iota ʻx\\ [*110·33] &=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\iota ʻx.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ [*73·45] &=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \in 1.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ [*52·1] &=\hat{\xi} \{(\exists \gamma ,\delta ,y).\gamma \text{ sm }\alpha .\delta =\iota ʻy.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ [*13·195.*51·211] &=\hat{\xi} \{(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}.\supset \vdash .\text{Prop} \end{array} \]

The above proposition is much used in the theory of finite and infinite, both cardinal and ordinal. It connects mathematical induction for inductive cardinals with mathematical induction for inductive classes (cf. *120).

*110·631. \(\vdash :\mu \in \text{NC}.\supset \mu +_{\text{c}}1=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \text{ sm }ʻʻ\mu .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}\)

Dem. \[ \begin{array}{l} \vdash .*110·211.*101·21.\supset \\ \vdash :\text{Hp}.\supset .\mu +_{\text{c}} 1&=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ1.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ [*101·28] &=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in 1.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ [*52·1.*51·211] &=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \text{ sm }ʻʻ\mu .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}:\supset \vdash .\text{Prop} \end{array} \]

The proposition \[ \mu \in \text{NC}.\supset .\mu +_{\text{c}} 1=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \mu .y{\sim}\in \gamma .\xi \text{ sm }\gamma \cup \iota ʻy\} \] which might at first sight seem demonstrable, will only be true universally if the total number of objects in any one type is not finite. For suppose \(\alpha\) is a type, and \(\mu=\text{N}_{0}\text{c}ʻ\alpha\). Then if \(\alpha\) is a finite class, \(\mu =\iota ʻ\alpha\). Hence \(\gamma \in \mu . \supset _{\gamma ,y}. y\in \gamma\). Hence \(\hat{\xi} \{(\exists \gamma ,y).\gamma \in \mu .y{\sim}\in \gamma.\xi \text{ sm }(\gamma \cup \iota ʻy)\}=\Lambda\) in all types. But \(\mu +_{\text{c}} 1\) will exist in all types higher than that of \(\gamma\). If on the other hand the number of entities in \(\alpha\) is infinite, we shall have \[ y\in \alpha .\supset .\alpha -\iota ʻy\in \text{Nc}ʻ\alpha .y{\sim}\in \alpha -\iota ʻy. \] Hence in this case the above proposition will be true universally.

[Pg 86]

*110·632. \(\vdash :\mu \in \text{NC}.\supset .\mu +_{\text{c}} 1=\hat{\xi} \{(\exists y).y\in \xi .\xi -\iota ʻy\in \text{ sm }ʻʻ\mu \}\)

Dem. \[ \begin{array}{l} \vdash .*110·631.*51·211·22.\supset \\ \vdash :\text{Hp}.&\supset .\mu +_{\text{c}} 1=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \text{ sm }ʻʻ\mu .y\in \xi .\gamma =\xi -\iota ʻy\}\\ [*13·195] &=\hat{\xi} \{(\exists y).y\in \xi .\xi -\iota ʻy\in \text{ sm }ʻʻ\mu\}:\supset \vdash .\text{Prop} \end{array} \]

*110·64. \(\vdash .0 +_{\text{c}} 0=0 \quad[*110·62]\)

*110·641. \(\vdash .1 +_{\text{c}} 0=0 +_{\text{c}} 1=1 \quad[*110·51·61.*101·2]\)

*110·642. \(\vdash .2 +_{\text{c}} 0=0 +_{\text{c}} 2=2 \quad[*110·51·61.*101·31]\)

*110·643. \(\vdash .1 +_{\text{c}} 1=2\)

Dem. \[ \begin{array}{l} \vdash .*110·632.*101·21·28.\supset \\ \vdash .1 +_{\text{c}} 1=\hat{\xi} \{(\exists y).y\in \xi .\xi -\iota ʻy\in 1\}\\ [*54·3] =2.\supset \vdash .\text{Prop} \end{array} \]

The above proposition is occasionally useful. It is used at least three times, in *113·66 and *120·123·472.

*110·7·71 are required for proving *110·72, and *110·72 is used in *117·3, which is a fundamental proposition in the theory of greater and less.

*110·7. \(\vdash :\beta \subset \alpha .\supset .(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu\)

Dem. \[ \begin{array}{l} \vdash .*24·411·21.\supset \vdash :\text{Hp}.&\supset .\alpha =\beta \cup (\alpha -\beta ).\beta \cap (\alpha -\beta )=\Lambda .\\ [*110·32] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ(\alpha -\beta ):\supset \vdash .\text{Prop} \end{array} \]

*110·71. \(\vdash :(\exists \mu ).\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu .\supset .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha\)

Dem. \[ \begin{array}{l} \vdash .*100·3.*110·4.\supset \\ \vdash :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu .&\supset .\mu \in \text{NC}-\iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash .*110·3.\supset \vdash :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} &\text{Nc}ʻ\gamma . \equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ(\beta + \gamma ).\\ [*100·3·31] &\supset .\alpha \text{ sm }(\beta + \gamma ).\\ [*73·1] &\supset .(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \cup \Lambda _{\beta }\downarrow ʻʻ\iotaʻʻ\gamma .\\ [*37·15] &\supset .(\exists R).R\in 1 \rightarrow 1.\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \subset \text{ᗡ}ʻR.Rʻʻ\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \subset \alpha .\\ [*110·12.*73·22] &\supset .(\exists \delta ).\delta \subset \alpha .\delta \text{ sm }\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 87]

The above proof depends upon the fact that "\(\text{Nc}ʻ\alpha\)" and "\(\text{Nc}ʻ\beta +_{\text{c}} \mu\)" are typically ambiguous, and therefore, when they are asserted to be equal, this must hold in any type, and therefore, in particular, in that type for which we have \(\alpha \in \text{Nc}ʻ\alpha\), i.e. for \(\text{N}_{0}\text{c}ʻ\alpha\). This is why the use of *100·3 is legitimate.

*110·72. \(\vdash :(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha .\equiv .(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu\)

Dem. \[ \begin{array}{l} \vdash .*100·321.*110·7.\supset \\ \vdash \colon\ldotp \delta \text{ sm } \beta .\delta \subset \alpha .&\supset :\text{Nc}ʻ\delta =\text{Nc}ʻ\beta :(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\delta +_{\text{c}} \mu :\\ [*13·12] &\supset :(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu &\qquad \text{(1)}\\ \vdash .(1).*110·71.\supset \vdash .\text{Prop} \end{array} \]

Summary of *111.

The arithmetical properties of a class, so far as these do not require or assume that it is a class of classes, are the same for any similar class. But a class of classes has many arithmetical properties which it does not share with all similar classes of classes. For example, if \(\kappa\) is a class of classes, the number of members of \(sʻ\kappa\) is an arithmetical property of \(\kappa\), but it is obvious that this is not determined by the number of members of \(\kappa\), but requires also a knowledge of the numbers of members of members of \(\kappa\). For example, let \(\kappa\) consist of the two members \(\alpha\) and \(\beta\), and let \(\lambda\) consist of \(\gamma\) and \(\delta\). Then \(\kappa \text{ sm } \lambda\); but in order to be able to infer \(sʻ\kappa \text{ sm } sʻ\lambda\), we require \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\) and \(\alpha \text{ sm } \gamma .\beta \text{ sm } \delta\) or \(\alpha \text{ sm } \delta .\beta \text{ sm } \gamma\) or some such further datum. The relation of "double similarity," to be defined in the present number, is a relation between classes of classes, which, when it holds between \(\kappa\) and \(\lambda\), insures that all the arithmetical properties of \(\kappa\) and \(\lambda\) are the same, e.g. we have (in particular) \(\text{Nc}ʻsʻ\kappa =\text{Nc}ʻsʻ\lambda\) and \(\text{Nc}ʻ{\in}_{\Delta }ʻ\kappa=\text{Nc}ʻ{\in}_{\Delta }ʻ\lambda\). This relation we denote by "\(\text{sm}\, \text{sm}\)," which is to be read as one symbol. It is defined as follows: We define first the class of "double correlators" of \(\kappa\) and \(\lambda\), which we denote by "\(\kappa \overline{\text{ sm }}\,\overline{\text{ sm }}\, \lambda\)," and of which the definition is

*111·01. \(\kappa \overline{\text{ sm }}\, \overline{\text{ sm }} \lambda =(1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻsʻ\lambda \cap \hat{T} (\kappa =T_{\in }ʻʻ\lambda ) \quad\text{Df}\)

so that \[ \vdash :T\in \kappa \overline{\text{ sm }}\, \overline{\text{ sm }} \lambda .\equiv .T\in 1\rightarrow 1.\text{ᗡ}ʻT=sʻ\lambda .\kappa =T_{\in }ʻʻ\lambda . \]

We then define "\(\kappa \text{ sm } \text{ sm } \lambda\)" as meaning that \(\kappa \overline{\text{ sm }}\, \overline{\text{ sm }} \lambda\) is not null, i.e. that there is at least one double correlator of \(\kappa\) and \(\lambda\).

To illustrate the nature of a double correlator, let us suppose that \(\kappa\) consists of the two classes \(\alpha _{1}\) and \(\alpha_{2}\), and that \(\alpha _{1}\) consists of \(x_{11}\), \(x_{12}\), while \(\alpha _{2}\) consists of \(x_{21}\), \(x_{22}\), \(x_{23}\). Similarly let \(\lambda\) consist of \(\beta _{1}\) and \(\beta _{2}\) while \(\beta _{1}\) consists of \(y_{11}\), \(y_{12}\) and \(\beta_{2}\) consists of \(y_{21}\), \(y_{22}\), \(y_{23}\). Now let \(T\) correlate each \(x\) with the \(y\) having the same two suffixes. Then \(T\) is a one-one, and its converse domain is \(sʻ\lambda\). Moreover \(T_{\in }ʻ\beta _{1}\) (which is \(Tʻʻ\beta _{1}\)) = \(\alpha_{1}\), and \(T_{\in }ʻ\beta _{2}=\alpha _{2}\), so that \(T_{\in}ʻʻ\lambda =\kappa\). Thus \(T\) is a double correlator according to the definition.

The essential characteristic of a double correlator \(T\) is that (1) \(T\) is a correlator of \(sʻ\kappa\) and \(sʻ\lambda\), (2) \(T_{\in }\upharpoonright \lambda\) is a correlator of \(\kappa\) and [Pg 89]\(\lambda\). If we write \(S\) in place of \(T_{\in }\upharpoonright \lambda\), then if \(\beta \in \lambda\), we have \(Sʻ\beta \in \kappa\); moreover \(T\upharpoonright \beta\) is a correlator of \(Sʻ\beta\) and \(\beta\). Thus \(\kappa\) and \(\lambda\) are similar classes of similar classes. They are not merely this, however, for we not only know that \(Sʻ\beta\) is similar to \(\beta\), but we know a particular correlator of \(Sʻ\beta\) and \(\beta\), namely \(T\upharpoonright \beta\). This is essential to the use of double similarity, as will appear shortly.

Let us consider the relation between \(\kappa\) and \(\lambda\) which consists in their being similar classes of similar classes. This means that there is a correlator \(S\) of \(\kappa\) and \(\lambda\), such that, if \(\beta \in \lambda\), \(Sʻ\beta\) is similar to \(\beta\). That is to say, we are to consider the hypothesis \[ (\exists S).S\in 1\rightarrow 1.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda .S \,\unicode{x2abd}\, \text{ sm } \] or, as it may be more briefly expressed, \[ \exists !\kappa \overline{\text{ sm }} \lambda \cap \text{Rl}ʻ\text{ sm }. \] Let us assume \(S\in \kappa \overline{\text{ sm }} \lambda \cap\text{ᗡ}ʻʻʻ\text{ sm }\). If we attempt to prove (say) that \(sʻ\kappa\) is similar to \(sʻ\lambda\), we find that we are forced to assume the multiplicative axiom, unless \(\kappa\) and \(\lambda\) are finite. This necessity arises as follows. Let us put \[ \text{Crp}(S)ʻ\beta =(Sʻ\beta )\,\overline{\text{ sm }}\, \beta , \] where "\(\text{Crp}\)" stands for "correspondence." Then we know that whenever \(\beta \in \lambda\), \(\text{Crp}(S)ʻ\beta\) is not null. Further it is easy to prove that, if \(\kappa\) and \(\lambda\) are classes of mutually exclusive classes, and if we can pick out one representative member of \(\text{Crp}(S)ʻ\beta\) for each value of \(\beta\) which is a member of \(\lambda\), then the relational sum of all these representative correlations gives us a correlator of \(sʻ\kappa\) and \(sʻ\lambda\). That is, we have \[ \vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.S\in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\supset .\dot{s} ʻ\text{D}ʻR\in (sʻ\kappa )\,\overline{\text{ sm }}\, (sʻ\lambda ). \]

But in order to infer hence \(sʻ\kappa \text{ sm } sʻ\lambda\), we need \(\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda\), i.e. we need to be able to pick out a particular correlator for each pair of similar classes \(Sʻ\beta\) and \(\beta\). This, however, cannot be done in general without assuming the multiplicative axiom. It follows that we must not define two classes as having double similarity when \(\exists!\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }\), but must give a definition which enables us to specify a particular correlator for each pair of similar classes. This is what is effected by the above definition of double correlators, where our \(S\) is given as of the form \(T_{\in }\upharpoonright \lambda\), where \(T\in 1\rightarrow 1.\text{ᗡ}ʻT =sʻ\lambda\). If the multiplicative axiom is assumed, but in general not otherwise, we have (*111·5) \[ \kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset :\kappa \text{ sm } \text{ sm } \lambda .\equiv .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }. \]

In the present number, we shall begin with various properties of double correlators. We prove (*111·11) that \(T\) is a double correlator of \(\kappa\) and \(\lambda\) when, and only when, \(T\) is a correlator of \(sʻ\kappa\) and \(sʻ\lambda\), and \(T_{\in}\upharpoonright \lambda\) is a correlator of \(\kappa\) and \(\lambda\). We prove (*111·112) that in the same hypothesis, \(T_{\in }\upharpoonright \lambda \in\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }\). We prove (*111·13) that \(1\upharpoonright sʻ\lambda\) is a double correlator of \(\lambda\) with itself; that (*111·131) if \(T\) is a double correlator of[Pg 90] \(\kappa\) and \(\lambda\), \(\breve{T}\) is a double correlator of \(\lambda\) and \(\kappa\); that (*111·132) if \(S\), \(T\) are double correlators of \(\kappa\) with \(\lambda\) and of \(\lambda\) with \(\mu\) respectively, \(S\mid T\) is a double correlator of \(\kappa\) with \(\mu\). Hence it follows (*111·45·451·452) that double similarity is reflexive, symmetrical, and transitive.

We then proceed (*111·2—·34) to consider \(\text{Crp}(S)ʻʻ\lambda\), where it is to be supposed that \(S\) is a correlator of \(Sʻʻ\lambda\) and \(\lambda\), and that \(Sʻ\beta\) is similar to \(\beta\) if \(\beta \in \lambda\). We prove

*111·32. \[\begin{aligned}\vdash :\lambda ,Sʻʻ\lambda \in \text{Cls}^{2} \text{excl}.&S\in 1\rightarrow 1.R\in {\in}_{\Delta }ʻ\text{Crp}(S)ʻʻ\lambda .M = \dot{s} ʻ\text{D}ʻR.\supset .\\ &M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .Sʻʻ\lambda = M_{\in }ʻʻ\lambda .S\upharpoonright \lambda = M_{\in }\upharpoonright \lambda\end{aligned}\]

Thus in the case supposed, \(M\) is a double correlator of \(Sʻʻ\lambda\) and \(\lambda\). Thus

*111·322. \[\begin{aligned}\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.S\in \kappa \,\overline{\text{ sm }}\, \lambda .R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .&M = \dot{s} ʻ\text{D}ʻR.\supset .\\ &M\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .S = M_{\in }\upharpoonright \lambda\end{aligned}\]

We then proceed (*111·4—·47) to various propositions on "\(\text{ sm }\,\text{ sm }\)," and finally (*111·5·51·53) state three propositions which assume the multiplicative axiom, namely

*111·5. If \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\), then \(\kappa \text{ sm } \text{ sm } \lambda . \equiv .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }\).

*111·51. In the same case, \(\exists !\kappa\,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }.\supset .sʻ\kappa\text{ sm } sʻ\lambda\), i.e. if \(\kappa\) and \(\lambda\) are similar classes of mutually exclusive similar classes, their sums are similar.

*111·53. In the same case, if \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\), \(\kappa \text{ sm } \text{ sm } \lambda\). Hence the multiplicative axiom implies that two classes of \(\mu\) mutually exclusive classes each of which has \(\nu\) terms, have the same number of terms in their sum.

*111·01. \(\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda = (1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻsʻ\lambda \cap \hat{T} (\kappa = T_{\in }ʻʻ\lambda ) \quad\text{Df}\)

*111·02. \(\text{Crp}(S)ʻ\beta = (Sʻ\beta ) \,\overline{\text{ sm }}\, \beta \quad \text{Df}\)

*111·03. \(\text{ sm } \text{ sm } = \hat{\kappa} \hat{\lambda} (\exists ! \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda ) \quad\text{Df}\)

*111·1. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv .T\in 1\rightarrow 1.\text{ᗡ}ʻT= sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda \quad [(*111·01)]\)

*111·11. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv .T\in (sʻ\kappa ).\,\overline{\text{ sm }}\, (sʻ\lambda ).T_{\in }\upharpoonright \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda \)

Dem. \[ \begin{array}{l} \vdash .*37·25.\text{Fact}.\supset \vdash :\text{ᗡ}ʻT = sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda .&\supset .\text{D}ʻT = Tʻʻsʻ\lambda .\kappa = T_{\in }ʻʻ\lambda .\\ [*40·38] & \supset .\text{D}ʻT = sʻTʻʻʻ\lambda .\kappa = T_{\in }ʻʻ\lambda .\\ [(*37·04)] & \supset .\text{D}ʻT=sʻ\kappa &\qquad \text{(1)}\\ \vdash .*72·451.*60·57.*35·65.\supset \\ \vdash :T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .&\supset .T_{\in }\upharpoonright \lambda \in 1\rightarrow 1.\lambda = \text{ᗡ}ʻ(T_{\in }\upharpoonright \lambda ) &\qquad \text{(2)}\\ \vdash .*37·401. &\supset \vdash :\kappa = T_{\in }ʻʻ\lambda . \equiv .\kappa = \text{D}ʻ(T_{\in }\upharpoonright \lambda ) &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*4·71.&\supset \vdash :T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda . \equiv .\\ &T\in 1\rightarrow 1.\text{D}ʻT = sʻ\kappa .\text{ᗡ}ʻT = sʻ\lambda .T_{\in }\upharpoonright \lambda \in 1\rightarrow 1.\text{D}ʻ(T_{\in }\upharpoonright \lambda ) = \kappa .\text{ᗡ}ʻ(T_{\in }\upharpoonright \lambda ) = \lambda &\qquad \text{(4)}\\ \vdash .(4).*111·1.*73·03.\supset \vdash .\text{Prop} \end{array} \]

[Pg 91]

*111·111. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T_{\in }\upharpoonright \lambda \,\unicode{x2abd}\, \text{ sm }\)

Dem. \[ \begin{array}{l} \vdash . *111·1. *60·57.\supset \vdash :\text{Hp}.&\supset .T\in 1\rightarrow 1.\lambda \subset \text{Cl}ʻ\text{ᗡ}ʻT.\\ [*73·5] &\supset .T_{\in }\upharpoonright \lambda \,\unicode{x2abd}\, \text{ sm }:\supset \vdash .\text{Prop} \end{array} \]

*111·112. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T_{\in }\upharpoonright \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm } \quad[*111·11·111]\)

The two following propositions are useful lemmas for the case when \(T\) is replaced (as it often is) by \(T\upharpoonright \alpha\).

*111·12. \(\vdash :sʻ\lambda \subset \alpha .\supset .(T\upharpoonright \alpha )_{\in }ʻʻ\lambda = T_{\in }ʻʻ\lambda .(T\upharpoonright \alpha )_{\in }\upharpoonright \lambda = T_{\in }\upharpoonright \lambda\)

Dem. \[ \begin{array}{l} \vdash . *37·101·421. &\supset \vdash :\beta \subset \alpha .\supset .(T\upharpoonright \alpha )_{\in }ʻ\beta = T_{\in }ʻ\beta &\qquad \text{(1)}\\ \vdash . *40·13. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \lambda .\supset .\beta \subset \alpha &\qquad \text{(2)}\\ \vdash .(1).(2). \supset \vdash \colon\ldotp \text{Hp}.&\supset :\beta \in \lambda .\supset .(T\upharpoonright \alpha )_{\in }ʻ\beta = T_{\in }ʻ\beta :\\ [*37·69.*35·71] &\supset :(T\upharpoonright \alpha )_{\in }ʻʻ\lambda = T_{\in }ʻʻ\lambda .(T\upharpoonright \alpha )_{\in }\upharpoonright \lambda = T_{\in }\upharpoonright \lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*111·121. \(\vdash .(T\upharpoonright sʻ\lambda )_{\in }ʻʻ\lambda = T_{\in }ʻʻ\lambda = (T_{\in }\upharpoonright \lambda )ʻʻ\lambda .(T\upharpoonright sʻ\lambda )_{\in }\upharpoonright \lambda = T_{\in }\upharpoonright \lambda\)

Dem. \[ \begin{array}{l} \vdash .*37·421.\supset \vdash .T_{\in }ʻʻ\lambda = (T_{\in }\upharpoonright \lambda )ʻʻ\lambda &\qquad \text{(1)}\\ \vdash .(1).*111·12 \frac{sʻ\lambda}{\alpha}.\supset \vdash .\text{Prop} \end{array} \]

*111·13. \(\vdash .I\upharpoonright sʻ\lambda \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda\)

Dem. \[ \begin{array}{l} \vdash .*72·17.*50·5·52.&\supset \vdash .I\upharpoonright sʻ\lambda \in 1\rightarrow 1.\text{ᗡ}ʻ(I\upharpoonright sʻ\lambda ) = sʻ\lambda &\qquad \text{(1)}\\ \vdash .*111·121. \supset \vdash .(I\upharpoonright sʻ\lambda )_{\in }ʻʻ\lambda &= I_{\in }ʻʻ\lambda \\ [*50·16·17] &= \lambda &\qquad \text{(2)}\\ \vdash .(1).(2).*111·1.\supset \vdash .\text{Prop} \end{array} \]

*111·131. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv .\breve{T} \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa\)

Dem. \[ \begin{array}{l} \vdash .*71·212.&\supset \vdash :T\in 1\rightarrow 1. \equiv .\breve{T} \in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash .*111·11.&\supset \vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .\text{D}ʻT = sʻ\kappa &\qquad \text{(2)}\\ \vdash .*111·1.(2).*60·57.\supset \\ \vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .&\supset .T\in 1\rightarrow 1.\kappa \subset \text{Cl}ʻ\text{D}ʻT.\lambda \subset \text{Cl}ʻ\text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda .\\ [*74·6] &\supset .\lambda = (\breve{T} )_{\in }ʻʻ\kappa &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*111·1.&\supset \vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .\breve{T} \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa &\qquad \text{(4)}\\ \vdash .(4)\frac{\breve{T}}{T}. &\supset \vdash :\breve{T} \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa .\supset .T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda &\qquad \text{(5)}\\ \vdash .(4).(5).\supset \vdash .\text{Prop} \end{array} \]

[Pg 92]

*111·132. \(\vdash : S \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . T \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \mu .\supset . S \mid T \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \mu\)

Dem. \[ \begin{array}{l} \vdash . *111·11 . *73·311 . \supset \\ \vdash : \text{Hp} . &\supset . S \mid T e (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\mu ) . (S_{\in } \upharpoonright \lambda ) \mid (T_{\in } \upharpoonright \mu ) \in \kappa \,\overline{\text{ sm }}\, \mu &\qquad \text{(1)}\\ \vdash . *35·354 . &\supset \vdash . (S_{\in } \upharpoonright \lambda ) \mid (T_{\in } \upharpoonright \mu ) = S_{\in } \mid (\lambda \upharpoonleft T_{\in } \upharpoonright \mu ) &\qquad \text{(2)}\\ \vdash . *74·251 . *111·1 . &\supset \vdash : \text{Hp} . \supset . S_{\in } \mid (\lambda \upharpoonleft T_{\in } \upharpoonright \mu ) = S_{\in } \mid (T_{\in } \upharpoonright \mu )\\ [*35·23] &= (S_{\in } \mid T_{\in }) \upharpoonright \mu \\ [*37·34] &= (S \mid T)_{\in } \upharpoonright \mu &\qquad \text{(3)}\\ \vdash . (1) . (2) . (3) . \supset \vdash : \text{Hp} . &\supset . S \mid T_{\in } (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\mu ) . (S \mid T)_{\in } \upharpoonright \mu \in \kappa \,\overline{\text{ sm }}\, \mu .\\ [*111·11] &\supset . S \mid T \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \mu : \supset \vdash . \text{Prop} \end{array} \]

*111·14. \(\vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv . T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . sʻ\lambda \subset \text{ᗡ}ʻT . \kappa = T_{\in }ʻʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash . *111·1·121 . \supset \\ \vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . &\equiv . T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . \text{ᗡ}ʻ(T \upharpoonright sʻ\lambda ) = sʻ\lambda . \kappa = T_{\in }ʻʻ\lambda .\\ [*35·65] &\equiv . T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . sʻ\lambda \subset \text{ᗡ}ʻT . \kappa = T_{\in }ʻʻ\lambda : \supset \vdash . \text{Prop} \end{array} \]

*111·15. \(\vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv . T \upharpoonright sʻ\lambda \in (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\lambda ) . T_{\in } \upharpoonright \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda\)

Dem. \[ \begin{array}{l} \vdash . *111·11 . \supset \\ \vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv . T \upharpoonright sʻ\lambda \in (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\lambda ) . (T \upharpoonright sʻ\lambda )_{\in } \upharpoonright \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda &\qquad \text{(1)}\\ \vdash . (1) . *111·121 . \supset \vdash . \text{Prop} \end{array} \]

*111·16. \(\vdash : \exists ! \alpha \,\overline{\text{ sm }}\, \beta \cap \gamma \,\overline{\text{ sm }}\, \delta . \supset . \alpha = \gamma . \beta = \delta\)

Dem. \[ \begin{array}{l} \vdash . *73·03 . \supset \vdash : \text{Hp} . &\supset . (\exists R) . \text{D}ʻR = \alpha . \text{ᗡ}ʻR = \beta . \text{D}ʻR = \gamma . \text{ᗡ}ʻR = \delta .\\ [*13·171] &\supset . \alpha = \gamma . \beta = \delta : \supset \vdash . \text{Prop} \end{array} \]

*111·18. \(\vdash . \alpha \,\overline{\text{ sm }}\, \beta \subset (\alpha \uparrow \beta )_{\Delta }ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash . *35·83 . *73·03 . &\supset \vdash : R \in \alpha \,\overline{\text{ sm }}\, \beta . \supset . R \,\unicode{x2abd}\, \alpha \uparrow \beta &\qquad \text{(1)}\\ \vdash . *73·03 . &\supset \vdash : R \in \alpha \,\overline{\text{ sm }}\, \beta . \supset . R \in 1 \rightarrow \text{Cls} . \text{ᗡ}ʻR = \beta &\qquad \text{(2)}\\ \vdash . (1) . (2) . *80·14 . &\supset \vdash . \text{Prop} \end{array} \]

The class \((\alpha \uparrow \beta )_{\Delta }ʻ\beta\) is important, being the class of Cantor's "Belegungen," used by him to define exponentiation; we have in fact \[ \text{Nc}ʻ(\alpha \uparrow \beta )_{\Delta }ʻ\beta = (\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta }. \] Thus the above proposition shows that \(\text{Nc}ʻ(\alpha\,\overline{\text{ sm }}\, \beta )\) is less than or equal to \((\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta}\); and since, whenever it is not zero, \(\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta\), it is less than or equal to \[ (\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\alpha }. \]

[Pg 93]

The following propositions lead up to *111·32·33·34:

*111·2. \(\vdash :\text{E}!Sʻ\beta .\supset .\text{Crp}(S)ʻ\beta = (Sʻ\beta )\,\overline{\text{ sm }}\, \beta \quad[*14·28.(*111·02)]\)

*111·201. \(\vdash :f\{\text{Crp}(S)ʻ\beta\}. \equiv .f\{(Sʻ\beta )\,\overline{\text{ sm }}\, \beta\} \quad[*4·2.(*111·02)]\)

*111·202. \[\begin{aligned}\vdash :R\in \text{Crp}(S)ʻ\beta . \equiv .R\in 1\rightarrow 1.\text{D}ʻR = &Sʻ\beta .\text{ᗡ}ʻR = \beta \\ &[*111·201.*73·03]\end{aligned}\]

*111·21. \(\vdash :\exists !\text{Crp}(S)ʻ\beta . \equiv .Sʻ\beta \text{ sm }\beta \quad[*111·201.*73·04]\)

*111·211. \(\vdash :\exists !\text{Crp}(S)ʻ\beta .\supset .\text{E}!Sʻ\beta .\beta \in \text{ᗡ}ʻS \quad[*111·21.*14·21.*33·43]\)

*111·22. \(\vdash \colon\ldotp \beta \in \text{ᗡ}ʻS.\supset _{\beta }.\exists !\text{Crp}(S)ʻ\beta : \equiv .S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }\)

Dem. \[ \begin{array}{l} \vdash .*111·21.\supset \vdash \colon\ldotp \beta \in \text{ᗡ}ʻS.\supset _{\beta }.\exists !\text{Crp}(S)ʻ\beta : &\equiv :\beta \in \text{ᗡ}ʻS.\supset _{\beta }.Sʻ\beta \text{ sm }\beta :\\ [*72·93] &\equiv :S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*111·221. \(\vdash \colon\ldotp S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }.\supset :\exists !\text{Crp}(S)ʻ\beta . \equiv .\beta \in \text{ᗡ}ʻS\)

Dem. \[ \begin{array}{l} \vdash .*111·22 .\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \text{ᗡ}ʻS.\supset .\exists !\text{Crp}(S)ʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*111·211.\supset \vdash .\text{Prop} \end{array} \]

*111·23. \(\vdash :S\in 1\rightarrow 1.\beta \in \text{ᗡ}ʻS.\supset .\text{Crp}(S)ʻ\beta = \text{Cnv}ʻʻ\text{Crp}(\breve{S} )ʻSʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*111·2.*71·163.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\text{Crp}(S)ʻ\beta & = (Sʻ\beta )\,\overline{\text{ sm }}\, \beta \\ [*73·301] &= \text{Cnv}ʻʻ(\beta \,\overline{\text{ sm }}\, Sʻ\beta )\\ [*72·241] &= \text{Cnv}ʻʻ(\breve{S} ʻSʻ\beta \,\overline{\text{ sm }}\, Sʻ\beta ) &\qquad \text{(1)}\\ \vdash .(1).*111·201 \frac{\breve{S},Sʻ\beta}{S,\, \beta}.\supset \vdash .\text{Prop} \end{array} \]

*111·24. \(\vdash :S\in 1\rightarrow \text{Cls}.\lambda \subset \text{ᗡ}ʻS.\supset .\text{Crp}(S)ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\)

Dem. \[ \begin{array}{l} \vdash .*111·2.*71·163.&\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\beta ,\gamma \in \lambda .\supset _{\beta , \gamma }.\text{Crp}(S)ʻ\beta = (Sʻ\beta )\,\overline{\text{ sm }}\, \beta .\text{Crp}(S)ʻ\gamma = (Sʻ\gamma )\,\overline{\text{ sm }}\, \gamma .&\qquad \text{(1)}\\ [*111·16] &\supset _{\beta , \gamma }.\exists !\text{Crp}(S)ʻ\beta \cap \text{Crp}(S)ʻ\gamma .\supset .\beta = \gamma .\\ [(1).*30·37] &\supset .\text{Crp}(S)ʻ\beta = \text{Crp}(S)ʻ\gamma &\qquad \text{(2)}\\ \vdash .(2).*37·63.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\rho ,\sigma \in \text{Crp} (S)ʻʻ\lambda .\exists !\rho \cap \sigma .\supset _{\rho ,\sigma }.\rho = \sigma \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*111·25. \[\begin{aligned}&\vdash :S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }.\lambda \subset \text{ᗡ}ʻS.\supset .\text{Crp}(S)ʻʻ\lambda \in \text{Cls ex}^{2}\,\text{excl}\\ &[*111·24·22]\end{aligned}\]

[Pg 94]

*111·3. \(\vdash : \lambda \in \text{Cls ex}^{2}\,\text{excl} . \supset . \dot{s} ʻʻ\text{D}ʻʻ {\in}_{\Delta} ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda \subset (\alpha \uparrow sʻ\lambda )_{\Delta }ʻsʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash . *37·29 . *24·12 . \supset \\ \vdash : {\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda = \Lambda . &\supset . \dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda \subset (\alpha \uparrow sʻ\lambda )_{\Delta }ʻsʻ\lambda &\qquad \text{(1)}\\ \vdash . *83·1 . \supset \\ \vdash \colon\ldotp \text{Hp} . \exists ! {\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda . \supset : \beta \in \lambda . &\supset _{\beta } . \exists ! \alpha \,\overline{\text{ sm }}\, ʻ\beta .\\ [*111·18] &\supset _{\beta } . \exists ! (\alpha \uparrow \beta )_{\Delta }ʻ\beta .\\ [*80·15] &\supset _{\beta } . \exists ! (\alpha \uparrow sʻ\lambda )_{\Delta }ʻ\beta :\\ [*80·83] &\supset : \{(\alpha \uparrow sʻ\lambda )_{\Delta }ʻʻ\lambda\} \upharpoonleft (\alpha \uparrow sʻ\lambda )_{\Delta } \in 1 \rightarrow 1 &\qquad \text{(2)}\\ \vdash . (2) . *111·18 . *85·72 \frac{\alpha \,\overline{\text{ sm }}\, , (\alpha \uparrow sʻ\lambda )_{\Delta }}{R,\, S}. \supset \\ \vdash : \text{Hp} . \exists ! {\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda . &\supset . \text{D}ʻʻ{\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda \subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\alpha \uparrow sʻ\lambda )_{\Delta }ʻʻ\lambda .\\ [*37·2] \supset . \dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda &\subset \dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ(\alpha \uparrow sʻ\lambda )_{\Delta }ʻʻ\lambda \\ [*85·27] &\subset (\alpha \uparrow sʻ\lambda )_{\Delta }ʻsʻ\lambda &\qquad \text{(3)}\\ \vdash . (1) . (3) . \supset \vdash . \text{Prop} \end{array} \]

*111·31. \[\begin{aligned}\vdash : \lambda , Sʻʻ\lambda \in \text{Cls}^{2} \text{excl} . S \in 1 \rightarrow 1 . R \in {\in}_{\Delta}ʻ\text{Crp} &(S)ʻʻ\lambda . \supset .\\ &\dot{s} ʻ\text{D}ʻR \in (sʻSʻʻ\lambda ) \,\overline{\text{ sm }}\, (sʻ\lambda )\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *83·2 . \supset \\ \vdash \colon\ldotp \text{Hp} . \supset : \beta \in \lambda . &\equiv . Rʻ\text{Crp} (S)ʻ\beta \in \text{Crp} (S)ʻ\beta .\\ [*111·202] &\equiv . Rʻ\text{Crp} (S)ʻ\beta \in 1 \rightarrow 1 . \text{D}ʻRʻ\text{Crp} (S)ʻ\beta = Sʻ\beta .\\ &\text{ᗡ}ʻRʻ\text{Crp} (S)ʻ\beta = \beta &\qquad \text{(1)}\\ \vdash . (1) . *72·322 . &\supset \vdash : \text{Hp} . \supset . \dot{s} ʻRʻʻ\text{Crp} (S)ʻʻ\lambda \in 1 \rightarrow 1 .\\ [*80·34] &\supset . \dot{s} ʻ\text{D}ʻR \in 1 \rightarrow 1 &\qquad \text{(2)}\\ \vdash . (1) . *37·68 . *50·17 . \supset \vdash : \text{Hp} . &\supset . \text{D}ʻʻRʻʻ\text{Crp} (S)ʻʻ\lambda = Sʻʻ\lambda .\\ &\text{ᗡ}ʻʻRʻʻ\text{Crp} (S)ʻʻ\lambda = \lambda .\\ [*80·34] &\supset . \text{D}ʻʻ\text{D}ʻR = Sʻʻ\lambda . \text{ᗡ}ʻʻ\text{D}ʻR = \lambda .\\ [*41·43·44] &\supset . \text{D}ʻ\dot{s} ʻ\text{D}ʻR = sʻSʻʻ\lambda . \text{ᗡ}ʻ\dot{s} ʻ\text{D}ʻR = sʻ\lambda &\qquad \text{(3)}\\ \vdash . (2) . (3) . *73·03 . \supset \vdash . \text{Prop} \end{array} \]

*111·311. \[\begin{aligned}&\vdash : \lambda , Sʻʻ\lambda \in \text{Cls}^{2} \text{excl} . S \in 1 \rightarrow 1 . \exists ! {\in}_{\Delta}ʻ\text{Crp} (S)ʻʻ\lambda . \supset · sʻSʻʻ\lambda \text{ sm } sʻ\lambda\\ &[*111·31 . *73·04]\end{aligned}\]

*111·313. \[\begin{aligned}\vdash : \lambda \in \text{Cls}^{2} \text{excl} . R \in &{\in}_{\Delta }ʻ\text{Crp} (S)ʻʻ\lambda . \beta \in \lambda . M = \dot{s} ʻ\text{D}ʻR . \supset .\\ &M \upharpoonright \beta = Rʻ\text{Crp} (S)ʻ\beta . M \upharpoonright \beta \in \text{Crp} (S)ʻ\beta\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *83·2 . \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \alpha \in \lambda . &\supset _{\alpha } : Rʻ\text{Crp} (S)ʻ\alpha \in \text{Crp} (S)ʻ\alpha : &\qquad \text{(1)}\\ [*111·202] &\supset _{\alpha } : \text{ᗡ}ʻRʻ\text{Crp} (S)ʻ\alpha = \alpha :\\ [*33·14 . *4·71] &\supset _{\alpha } : x {Rʻ\text{Crp} (S)ʻ\alpha } y . \equiv . x {Rʻ\text{Crp} (S)ʻ\alpha } y . y \in \alpha &\qquad \text{(2)}\\ \vdash .*35·101.*83·23.*41·11.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :x(M\upharpoonright \beta )y.&\equiv .(\exists \alpha ).\alpha \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\alpha \}y.y\in \beta \\ [(2)] & \equiv .(\exists \alpha ).\alpha \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\alpha\}y.y\in \alpha \cap \beta .\\ [*84·11.*22·5] &\equiv .(\exists \alpha ).\alpha \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\alpha\}y.y\in \beta .\alpha = \beta .\\ [*13·195] &\equiv .\beta \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\beta \}y.y\in \beta .\\ [\text{Hp}.*4·73.(2)] &\equiv .x\{Rʻ\text{Crp}(S)ʻ\beta\}y &\qquad \text{(3)}\\ \vdash .(1).(3).\supset \vdash .\text{Prop} \end{array} \]

[Pg 95]

*111·32. \[\begin{aligned}\vdash :\lambda ,Sʻʻ\lambda \in &\text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .M = \dot{s} ʻ\text{D}ʻR.\supset .\\ &M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .Sʻʻ\lambda = M_{\in }ʻʻ\lambda .S\upharpoonright \lambda = M_{\in }\upharpoonright \lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*111·31.*73·03.\supset \vdash :\text{Hp}.\supset .M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda &\qquad \text{(1)}\\ \vdash .*111·313·202. \supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \lambda .\supset .\text{D}ʻ(M\upharpoonright \beta ) = Sʻ\beta .\text{ᗡ}ʻ(M\upharpoonright \beta ) = \beta .\\ [*37·25] \supset .(M\upharpoonright \beta )ʻʻ\beta = Sʻ\beta .\\ [*37·421·11] \supset .M_{\in }ʻ\beta = Sʻ\beta :\\ [*35·71.*37·69] \supset :M_{\in }\upharpoonright \lambda = S\upharpoonright \lambda .M_{\in }ʻʻ\lambda = Sʻʻ\lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*111·321. \[\begin{aligned}\vdash :\lambda ,&Sʻʻ\lambda \in \text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\supset .\\ &(\exists M).M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .Sʻʻ\lambda = M_{\in }ʻʻ\lambda .S\upharpoonright \lambda = M_{\in }\upharpoonright \lambda\\ &[*111·32]\end{aligned}\]

*111·322. \[\begin{aligned}\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.&S\in \kappa \,\overline{\text{ sm }}\, \lambda .R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .M = \dot{s} ʻ\text{D}ʻR.\supset .\\ &M\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .S = M_{\in }\upharpoonright \lambda \quad[*111·32·1.*35·66.*73·03]\end{aligned}\]

*111·33. \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax}.\supset :S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.&\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\kappa =Sʻʻ\lambda .\lambda \subset \text{ᗡ}ʻS.\supset .\\ &sʻ\kappa \text{ sm }sʻ\lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*111·221.\supset \\ \vdash \colon\ldotp S\in 1\rightarrow 1.&S\,\unicode{x2abd}\, \text{ sm }.\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\kappa =Sʻʻ\lambda .\lambda \subset \text{ᗡ}ʻS.\supset :\\ &\beta \in \lambda .\supset _{\beta }.\exists !\text{Crp}(S)ʻ\beta :\\ [*88·37] \supset :\text{Mult ax}.&\supset .\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\\ [*111·311] &\supset .sʻ\kappa \text{ sm }sʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*111·34. \[\begin{aligned}\vdash \colon\ldotp &\text{Mult ax}.\supset :\\ &(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS = \lambda .\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset .\\ &(\exists M).M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .\kappa = M_{\in }ʻʻ\lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*111·25.\supset \\ \vdash \colon\ldotp S\in 1\rightarrow 1.&S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset :\\ &\text{Crp}(S)ʻʻ\lambda \in \text{Cls ex}^{2}\,\text{excl}:\\ [*88·32] &\supset :\text{Mult ax}.\supset .\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\\ [*111·321] &\supset .(\exists M).M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .\kappa = M_{\in }ʻʻ\lambda &\qquad \text{(1)}\\ \vdash .(1).*10·11·23.\text{Comm}.\supset \vdash .\text{Prop} \end{array} \]

[Pg 96]

The following propositions are concerned with the elementary properties of "\(\text{sm} \,\text{sm}\)." It will be seen that they are closely analogous to those of "\(\text{sm}\)."

*111·4. \[\begin{aligned}&\vdash :\kappa \text{ sm } \text{ sm }\lambda . \equiv .(\exists T).T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda . \equiv .\exists !\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda \\ &[*111·1.(*111·03)]\end{aligned}\]

*111·401. \(\vdash :\kappa \text{ sm } \text{ sm } \lambda . \equiv .(\exists T).T\in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash . *22·42 . *111·4.\supset \vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .(\exists T).T\in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda &\qquad \text{(1)}\\ \vdash .(1).*111·14.\supset \vdash .\text{Prop} \end{array} \]

*111·402. \[\begin{aligned}&\vdash :\kappa \text{ sm } \text{ sm } \lambda . \equiv .(\exists T).T\upharpoonright sʻ\lambda \in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda\\ &[*111·14·1·121]\end{aligned}\]

*111·43. \[\begin{aligned}&\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda \\ &[*111·11·111]\end{aligned}\]

*111·44. \(\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .\kappa \text{ sm } \lambda .sʻ\kappa \text{ sm } sʻ\lambda \quad[*111·11·4.*73·03]\)

*111·45. \(\vdash .\lambda \text{ sm } \text{ sm } \lambda \quad[*111·13·4]\)

*111·451. \(\vdash :\kappa \text{ sm } \text{ sm } \lambda . \equiv .\lambda \text{ sm } \text{ sm } \kappa \quad[*111·131·4]\)

*111·452. \(\vdash :\kappa \text{ sm } \text{ sm } \lambda .\lambda \text{ sm } \text{ sm } \mu .\supset .\kappa \text{ sm } \text{ sm } \mu \quad[*111·132·4]\)

*111·46. \[\begin{aligned}&\vdash :\lambda ,Sʻʻ\lambda \in \text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\supset .Sʻʻ\lambda \text{ sm } \text{ sm } \lambda \\ &[*111·32·4]\end{aligned}\]

*111·47. \(\vdash \colon\ldotp \kappa \text{ sm } \text{ sm } \lambda .\supset :\kappa \in \text{Cls}^{2} \text{excl}. \equiv .\lambda \in \text{Cls}^{2} \text{excl}\)

Dem. \[ \begin{array}{l} \vdash . *111·4. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists T).T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .\kappa = Tʻʻʻ\lambda :\\ [*84·53] &\supset :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ \vdash .(1).*111·451.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\lambda \in \text{Cls}^{2} \text{excl} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*111·5. \[\begin{aligned}&\vdash \colon\colon \text{Mult ax}.\supset \colon\ldotp \kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset :\\ &\kappa \text{ sm } \text{ sm } \lambda . \equiv .(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\\ &\equiv .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm } \quad[*111·34·43·4]\end{aligned}\]

*111·51. \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.\supset .sʻ\kappa \text{ sm }sʻ\lambda \\ &[*111·5·44]\end{aligned}\]

*111·52.\(\vdash :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }\)

Dem. \[ \begin{array}{l} \vdash .*100·5.*73·1.&\supset \vdash :\text{Hp}.\supset .(\exists S).S\in 1\rightarrow 1.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda &\qquad \text{(1)}\\ \vdash .*100·5. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \kappa .\beta \in \lambda .\supset .\alpha \text{ sm } \beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*111·53. \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl excl}ʻ\nu .\supset .\kappa \text{ sm } \text{ sm } \lambda\\ &[*111·52·5]\end{aligned}\]

Summary of *112.

In this number, we return to the arithmetical operations. The definition of addition in *110 was only applicable to a finite number of summands, because the summands had to be enumerated. In the present number, we define the arithmetical sum of a class of classes, so that the summands are given as the members of a class, and do not require to be enumerated. Hence the definition in this number is as applicable to an infinite number of summands as to a finite number.

If \(\kappa\) is a class of mutually exclusive classes, the number of \(sʻ\kappa\) will be the sum of the numbers of members of \(\kappa\); i.e. if we write "\(\Sigma \text{Nc}ʻ\kappa\)" for the sum of the numbers of members of \(\kappa\), \[ \kappa \in \text{Cls}^{2} \text{excl}.\supset .\text{Nc}ʻsʻ\kappa =\Sigma \text{Nc}ʻ\kappa . \] But when the members of \(\kappa\) are not mutually exclusive, a term \(x\) which is a member of two members (say \(\alpha\) and \(\beta\)) of \(\kappa\) has to be counted twice over in obtaining the arithmetical sum of \(\kappa\), whereas in the logical sum \(x\) is only counted once. Thus we need a construction which shall duplicate \(x\), taking it first as a member of \(\alpha\), and then as a member of \(\beta\). This is effected if we replace \(x\) first by \(x\downarrow \alpha\), and then by \(x\downarrow \beta\). In fact, \(x\downarrow \alpha\) has the kind of arithmetical properties which we mean to secure when we speak of "\(x\) considered as a member of \(\alpha\)"—a phrase which, as it stands, does not serve our purpose, for \(x\) is simply \(x\) however we may choose to consider it. Thus we replace \(\alpha\) by \(\downarrow \alphaʻʻ\alpha\) and \(\beta\) by \(\downarrow \betaʻʻ\beta\) and so on; i.e. (using *85·5), we replace \(\alpha\) by \(\in \unicode{x21A7}\alpha\) and \(\beta\) by \(\in \unicode{x21A7}\beta\) and so on. These new classes are similar to \(\alpha\) and \(\beta\) and so on, and are mutually exclusive. Hence their logical sum has the number of terms which is wanted for the arithmetical sum of the members of \(\kappa\). Thus we put \[ \begin{aligned} \Sigma ʻ\kappa &=sʻ\in \unicode{x21A7}ʻʻ\kappa &\quad\text{Df},\\ \Sigma \text{Nc}ʻ\kappa &=\text{Nc}ʻ\Sigma ʻ\kappa &\quad\text{Df}. \end{aligned} \]

With regard to the second of these definitions, it is to be observed that \(\Sigma \text{Nc}ʻ\kappa\) is not a function of \(\text{Nc}ʻʻ\kappa\), unless no two members of \(\kappa\) are similar; for \(\text{Nc}ʻʻ\kappa\) cannot contain the same number twice over. For the same reason, if \(\lambda\) is a class of cardinals, and we define "\(\text{Sum}ʻ\lambda\)," we do not get what[Pg 98] is wanted for arithmetical addition, because our definition will not enable us to deal with summations in which there are numbers that are repeated. We could, if it were worth while, define "\(\text{Sum}ʻ\lambda\)" as follows: Take a class of classes \(\kappa\), consisting of one class having each number which is a member of \(\lambda\), i.e. let \(\kappa\) be a selection from \(\lambda\); then \(\Sigma ʻ\kappa\) will have the required number of terms. i.e. we might put \[ \text{Sum}ʻ\lambda = \hat{\xi} \{(\exists \kappa ).\kappa \in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda .\xi \text{ sm }\Sigma ʻ\kappa\} \quad\text{Df}. \] But since this definition is only available for sums in which no number is repeated, it is not worth while to introduce it.

In this number we prove the following propositions among others.

*112·15. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .sʻ\kappa \in \Sigma \text{Nc}ʻ\kappa\)

This is an extension of *110·32.

*112·17. \(\vdash :\kappa \text{ sm }\text{ sm }\lambda .\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda .\Sigma ʻ\kappa \text{ sm }\Sigma ʻ\lambda\)

The chief point in the above proposition is that it does not require \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\).

*112·2—·24 are concerned with the use of the multiplicative axiom and the propositions of *111 in which it appears as hypothesis. We have

*112·22. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\exists !(\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\cap \text{Rl}ʻ\text{ sm }.\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\)

whence we derive the proposition

*112·24. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\)

I.e. assuming the multiplicative axiom, two classes which each consist of \(\mu\) classes of \(\nu\) terms each have the same number of terms in their sum. This number would naturally be defined as \(\mu\) multiplied by \(\nu\), but owing to the necessity of the multiplicative axiom in this proposition, we have selected a different definition of multiplication (*113) which does not depend upon the multiplicative axiom. The reader should observe that the similarity of two classes, each of which consists of \(\mu\) mutually exclusive sets of \(\nu\) terms, cannot be proved in general without the multiplicative axiom.

The remaining propositions of this number give properties of \(\Sigma\) in special cases. We prove that \(\Sigma ʻ\Lambda = \Lambda\) (*112·3), that \(\Sigma \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha\) (*112·321), that \(\alpha \neq \beta .\supset .\Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup\iota ʻ\beta ) = \text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta\) (*112·34), which connects the definition of addition in this number with that in *110. Finally we prove the general associative law for addition, in the following two forms:

*112·41. \(\vdash .sʻ\Sigma ʻʻ\lambda = \Sigma ʻsʻ\lambda\)

*112·43. \(\vdash :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\text{Nc}ʻ\Sigma ʻ\Sigma ʻʻ\lambda = \text{Nc}ʻ\Sigma ʻsʻ\lambda\)

[Pg 99]

*112·01. \(\Sigma ʻ\kappa = sʻ\in \unicode{x21A7} ʻʻ\kappa \quad\text{Df}\)

*112·02. \(\Sigma \text{Nc}ʻ\kappa = \text{Nc}ʻ\Sigma ʻ\kappa \quad\text{Df}\)

*112·1. \(\vdash .\Sigma ʻ\kappa = sʻ\in \unicode{x21A7} ʻʻ\kappa \quad[*20·2. (*112·01)]\)

*112·101. \(\vdash .\Sigma \text{Nc}ʻ\kappa = \text{Nc}ʻ\Sigma ʻ\kappa = \text{Nc}ʻsʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*20·2. *112·1. (*112*02)]\)

*112·102. \(\vdash .\Sigma ʻ\kappa = \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .x\in \alpha .R=x\downarrow \alpha\}\)

Dem. \[ \begin{array}{l} \vdash . *85·6. *40·11. *112·1.\supset \\ \vdash . \Sigma ʻ\kappa &= \hat{R} \{(\exists \mu ,\alpha ).\alpha \in \kappa .\mu = \downarrow \alpha ʻʻ\alpha .R\in \mu\}\\ [*13·195] &= \hat{R} \{(\exists \alpha ).\alpha \in \kappa .R\in \downarrow \alpha ʻʻ\alpha\}\\ [*55·231] &= \hat{R} \{(\exists \alpha , x).\alpha \in \kappa .x\in \alpha .R=x\downarrow \alpha\}.\supset \vdash .\text{Prop} \end{array} \]

*112·103. \(\vdash .\Sigma ʻ\kappa = sʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \kappa .\mu =\downarrow \alpha ʻʻ\alpha\} \quad[*112·1. *85·6]\)

*112·11. \(\vdash :\beta \in \Sigma \text{Nc}ʻ\kappa .\equiv . \beta \text{ sm } sʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*112·101]\)

*112·12. \(\vdash .sʻ\in \unicode{x21A7}ʻʻ\kappa \in \Sigma \text{Nc}ʻ\kappa \quad[*112·11]\)

*112·13. \(\vdash :\lambda \text{ sm } \text{ sm } \in \unicode{x21A7}ʻʻ\kappa . \supset . sʻ\lambda \in \Sigma \text{Nc}ʻ\kappa \quad[*111·44. *112·11]\)

*112·14. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}. \supset . \in \unicode{x21A7} ʻʻ\kappa \text{ sm } \text{ sm } \kappa\)

Dem. \[ \begin{array}{l} \vdash . *21·33 .&\supset \vdash \colon\ldotp \text{Hp}.T= \hat{R} \hat{x} \{(\exists \alpha ).\alpha \in \kappa .x\in \alpha .R= x\downarrow \alpha\}.\supset :\\ &xTR.yTR . \supset . (\exists \alpha ,\beta ).R = x\downarrow \alpha .R = y\downarrow \beta .\\ [*55·31] &\supset .x=y:\\ [*71·17] &\supset :T\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash . *21·33 .\supset\\ \vdash :\text{Hp}(1).xTR.xTS. &\supset . (\exists \alpha ,\beta ).\alpha ,\beta \in \kappa .x\in \alpha \cap \beta .R = x\downarrow \alpha .S=x\downarrow \beta .\\ [*84·11.\text{Hp}] &\supset . (\exists \alpha ,\beta ).\alpha =\beta .R = x\downarrow \alpha .S = x\downarrow \beta .\\ [*13·195] &\supset .R=S:\\ [*71·171] &\supset :T\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ \vdash .*33·131.\supset \vdash \colon\ldotp \text{Hp}(1).\supset :x\in \text{ᗡ}ʻT. &\equiv . (\exists R,\alpha ).\alpha \in \kappa .x\in \alpha .R = x\downarrow \alpha .\\ [*55·12] &\equiv .x\in sʻ\kappa &\qquad \text{(3)}\\ \vdash .*37·1·11 .\supset \\ \vdash \colon\colon \text{Hp} .\supset \colon\ldotp \alpha \in \kappa .\supset : R\in T_{\in }ʻ\alpha . &\equiv . (\exists x,\beta ).x\in \alpha \cap \beta .\beta \in \kappa .R=x\downarrow \beta .\\ [*84·11.\text{Hp}] &\equiv . (\exists x,\beta ).x\in \alpha \cap \beta .\beta \in \kappa .\alpha =\beta .R = x\downarrow \beta .\\ [*13·195] &\equiv . (\exists x).x\in \alpha .R = x\downarrow \beta .\\ [*85·601] &\equiv .R\in \in \unicode{x21A7}ʻ\alpha \colon\ldotp \\ [*37·69] &\supset \colon\ldotp T_{\in }ʻʻ\kappa = \in \unicode{x21A7}ʻʻ\kappa &\qquad \text{(4)}\\ \vdash .(1).(2).(3).(4).*111·4.\supset \vdash .\text{Prop} \end{array} \]

[Pg 100]

*112·15. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}. \supset . sʻ\kappa \in \Sigma \text{Nc}ʻ\kappa \quad[*112·14·11. *111·44]\)

*112·151. \(sʻ\in \unicode{x21A7}ʻʻ\lambda = \hat{R} \{(\exists \alpha ,x).\alpha \in \lambda .x\in \alpha .R = x\downarrow \alpha \}.\dot{s} ʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \in \upharpoonright \lambda\)

Dem. \[ \begin{array}{l} \vdash .*40·11.(*85·5).\supset \\ \vdash . sʻ\in \unicode{x21A7}ʻʻ\lambda &= \hat{R} \{(\exists \alpha ).\alpha \in \lambda .R\in \downarrow \alpha ʻʻ\alpha\}\\ [*38·131] &= \hat{R} \{(\exists \alpha ,x).\alpha \in \lambda .x\in \alpha .R = x\downarrow \alpha\} &\qquad \text{(1)}\\ \vdash .(1).*41·11.\supset \\ \vdash .\dot{s} ʻsʻ\in \unicode{x21A7}ʻʻ\lambda &= \hat{y} \hat{\beta} \{(\exists R,\alpha ,x).\alpha \in \lambda .x\in \alpha .R = x\downarrow \alpha .yR\beta\}\\ [*13·195.*55·13] &= \hat{y} \hat{\beta} \{(\exists \alpha ,x).\alpha \in \lambda .x\in \alpha .y = x.\beta = \alpha\}\\ [*13·22] &= \hat{y} \hat{\beta} \{\beta \in \lambda .y\in \beta\}\\ [*35·101] &= \in \upharpoonright \lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

The following proposition is a lemma for *112·153, which is required for *112·16. *112·16 in turn is used in *112·17, which is a fundamental proposition in the theory of addition.

*112·152. \(\vdash :T\in 1\rightarrow Cls.\beta \subset \text{ᗡ}ʻT.\supset .(T\parallel \breve{T} _{\in })ʻʻ\in \unicode{x21A7}\beta = \in \unicode{x21A7}(Tʻʻ\beta )\)

Dem. \[ \begin{array}{l} \vdash . *37·6.*85·601.&\supset \vdash .(T\parallel \breve{T} _{\in })ʻʻ\in \unicode{x21A7}\beta = \hat{R} \{(\exists y).y\in \beta .R = (T\parallel \breve{T} _{\in })ʻ(y\downarrow \beta )\} &\qquad \text{(1)}\\ \vdash .(1).*55·61.\supset \\ \vdash :\text{Hp}.\supset .(T\parallel \breve{T} _{\in })ʻʻ\in \unicode{x21A7}\beta &= \hat{R} \{(\exists y).y\in \beta .R = (Tʻy)\downarrow (T_{\in }ʻ\beta )\}\\ [*37·11] &= \hat{R} \{(\exists y).y\in \beta .R = (Tʻy)\downarrow (Tʻʻ\beta )\}\\ [*38·131] &= \downarrow (Tʻʻ\beta )ʻʻ(Tʻʻ\beta )\\ [*85·601] &= \in \unicode{x21A7}(Tʻʻ\beta ):\supset \vdash .\text{Prop} \end{array} \]

In the following proposition, we have a double correlator of a sort which will frequently occur in cardinal arithmetic, namely \(T\parallel\breve{T}_{\in }\) with its converse domain limited, where \(T\) is a given double correlator (or single correlator, on other occasions). As appears from the propositions used in the above proof of *112·152, if \(T\) is a correlator whose converse domain includes \(\beta\) and has \(y\) as a member, \((T\parallel \breve{T} _{\in })ʻ(y\downarrow \beta ) =(Tʻy)\downarrow (Tʻʻ\beta )\). Thus \(T\parallel \breve{T}_{\in }\) is an operation which, when operating on suitable relations of individuals to classes (including selectors), turns the individuals into their correlates and the classes into the classes of their members' correlates. This is why it is a useful relation.

*112·153. \(T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright sʻ\in \unicode{x21A7}ʻʻ\lambda \in (\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\)

Dem. \[ \begin{array}{l} \vdash .*112·151.*41·43·44.&\supset \vdash .sʻ\text{D}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \text{D}ʻ(\in \upharpoonright \lambda ).sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \text{ᗡ}ʻ(\in \upharpoonright \lambda ).\\ [*62·41·43] &\supset \vdash .sʻ\text{D}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = sʻ\lambda .sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \lambda -\iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash . (1) . *111·1 . *37·231 . &\supset \vdash : \text{Hp} . \supset . sʻ\text{D}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \subset \text{ᗡ}ʻT . sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \subset \text{ᗡ}ʻT_{\in } &\qquad \text{(2)}\\ \vdash . *111·1 .*71·29 . &\supset \vdash : \text{Hp} . \supset . T \upharpoonright sʻ\text{D}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \in 1 \rightarrow 1 &\qquad \text{(3)}\\ \vdash . *111·11 . (1) . &\supset \vdash : \text{Hp} . \supset . T_{\in } \upharpoonright sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \in 1 \rightarrow 1 &\qquad \text{(4)}\\ \vdash . (2) . (3) . (4) . *74·775 \frac{sʻ\in \unicode{x21A7} ʻʻ\lambda , T, T_{\in }}{\lambda,\, Q,\, R}. &\supset \vdash : \text{Hp} . \supset . (T \parallel \breve{T} _{\in }) \upharpoonright sʻ\in \unicode{x21A7} ʻʻ\lambda \in 1 \rightarrow 1 &\qquad \text{(5)}\\ \vdash . *43·302 . &\supset \vdash . sʻ\in \unicode{x21A7} ʻʻ\lambda \subset \text{ᗡ}ʻ(T \parallel \breve{T} _{\in }) &\qquad \text{(6)}\\ \vdash . *112·152 . \supset \vdash : \text{Hp} . &\supset . (T \parallel \breve{T} _{\in })ʻʻʻ\in \unicode{x21A7} ʻʻ\lambda = \in \unicode{x21A7} ʻʻTʻʻʻ\lambda .\\ [*37·11] \supset . (T \parallel \breve{T} _{\in })_{\in }ʻʻ\in \unicode{x21A7} ʻʻ\lambda &= \in \unicode{x21A7} ʻʻT_{\in }ʻʻ\lambda \\ [*111·1 . \text{Hp}] &= \in \unicode{x21A7} ʻʻ\kappa &\qquad \text{(7)}\\ \vdash . (5) . (6) . (7) . *111·14 . \supset \vdash . \text{Prop} \end{array} \]

[Pg 101]

*112·16. \(\vdash : \kappa \text{ sm } \text{ sm } \lambda . \supset . \in \unicode{x21A7} ʻʻ\kappa \text{ sm } \text{ sm } \in \unicode{x21A7} ʻʻ\lambda \quad[*112·153 . *111·4]\)

*112·17. \(\vdash : \kappa \text{ sm } \text{ sm } \lambda . \supset . \Sigma \text{Nc}ʻ\kappa . \Sigma \text{Nc}ʻ\lambda . \Sigma ʻ\kappa \text{ sm } \Sigma ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash . *112·16 . *111·44 . \supset \vdash : \text{Hp} . \supset . sʻ\in \unicode{x21A7} ʻʻ\kappa \text{ sm } sʻ\in \unicode{x21A7} ʻʻ\lambda &\qquad \text{(1)}\\ \vdash . (1) . *112·1·101 . \supset \vdash . \text{Prop} \end{array} \]

*112·18. \(\vdash . \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\in \unicode{x21A7} ʻʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash . *85·61 . *112·15 . \supset \vdash . sʻ\in \unicode{x21A7} ʻʻ\kappa \in \Sigma \text{Nc}ʻ\in \unicode{x21A7} ʻʻ\kappa &\qquad \text{(1)}\\ \vdash . (1) . *112·12 . *100·34 . \supset \vdash . \text{Prop} \end{array} \]

*112·2. \[\begin{aligned}\vdash : S \in 1 \rightarrow 1 . \text{D}ʻS = \in &\unicode{x21A7} ʻʻ\kappa . \text{ᗡ}ʻS = \in \unicode{x21A7} ʻʻ\lambda . \exists ! {\in}_{\Delta} ʻ\text{Crp} (S)ʻʻ\lambda .\\ &\supset . \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda . \Sigma ʻ\kappa \text{ sm } \Sigma ʻ\lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *111·311 . *85·61 . \supset \vdash : \text{Hp} . \supset . sʻ\in \unicode{x21A7} ʻʻ\kappa \text{ sm } sʻ\in \unicode{x21A7} ʻʻ\lambda &\qquad \text{(1)}\\ \vdash . (1) . *112·1·101 . \supset \vdash . \text{Prop} \end{array} \]

*112·21. \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} . \supset : (\exists S) . S \in 1 \rightarrow 1 . &S \,\unicode{x2abd}\, \text{ sm } . \text{D}ʻS = \in \unicode{x21A7} ʻʻ\kappa . \text{ᗡ}ʻS = \in \unicode{x21A7} ʻʻ\lambda .\\ &\equiv . \in \unicode{x21A7} ʻʻ\kappa \text{ sm } \text{ sm } \in \unicode{x21A7} ʻʻ\lambda \quad[*111·5 . *85·61]\end{aligned}\]

*112·22. \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} . \supset : \exists ! (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, (\in \unicode{x21A7} ʻʻ\lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset .\\ \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda \quad[*112·17·18·21]\end{aligned}\]

*112·23. \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} . \supset : \kappa , \lambda \in \text{Cls}^{2} \text{excl} . \exists ! \kappa &\,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm } . \supset .\\ &sʻ\kappa , sʻ\lambda \in \Sigma \text{Nc}ʻ\kappa . \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *112·15 . &\supset \vdash : \text{Hp} . \kappa , \lambda \in \text{Cls}^{2} \text{excl} . \supset . sʻ\kappa \in \Sigma \text{Nc}ʻ\kappa . sʻ\lambda \in \Sigma \text{Nc}ʻ\lambda &\qquad \text{(1)}\\ \vdash . *111·51 . &\supset \vdash : \text{Hp} (1) . \exists ! \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset . sʻ\kappa \text{ sm } sʻ\lambda &\qquad \text{(2)}\\ \vdash . (1) . (2) . &\supset \vdash . \text{Prop} \end{array} \]

*112·231. \(\vdash : S \in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \in (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, ( \in \unicode{x21A7} ʻʻ\lambda ) \cap \text{Rl} ʻ\text{ sm }\)

Dem. \[ \begin{array}{l} \vdash . *73·63 . *85·601 . &\supset \vdash : S \in \kappa \,\overline{\text{ sm }}\, \lambda . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \in (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, (\in \unicode{x21A7} ʻʻ\lambda ) &\qquad \text{(1)}\\ \vdash . *85·601 . *73·33·34 .& \supset \vdash : S \,\unicode{x2abd}\, \text{ sm } . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \,\unicode{x2abd}\, \text{ sm } &\qquad \text{(2)}\\ \vdash . (1) . (2) . &\supset \vdash : S \in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \in (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, (\in \unicode{x21A7} ʻʻ\lambda ) \cap \text{Rl} ʻ\text{ sm }:\\ &\supset \vdash . \text{Prop} \end{array} \]

[Pg 102]

*112·24. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*111·52.\supset \vdash :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .&\supset .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.\\ [*112·231] &\supset .\exists !(\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\cap \text{ᗡ}ʻʻʻ\text{ sm } &\qquad \text{(1)}\\ \vdash .(1).*111·51.*85·61.\supset \\ \vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .&\supset .sʻ\in \unicode{x21A7}ʻʻ\kappa \text{ sm }sʻ\in \unicode{x21A7}ʻʻ\lambda .\\ [*112·101] &\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*112·3. \(\vdash .\Sigma ʻ\Lambda = \Lambda \quad[*37·29.*40·21.*112·1]\)

*112·301. \(\vdash .\Sigma ʻ\iota ʻ\Lambda = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*112·102.\supset \vdash .\Sigma ʻ\iota ʻ\Lambda &= \hat{R} \{(\exists \alpha ,x).\alpha \in \iota ʻ\Lambda .x\in \alpha .R = x\downarrow \alpha\}\\ [*51·15] &= \hat{R} \{(\exists x).x\in \Lambda .R = x\downarrow \Lambda\}\\ [*24·15] &= \Lambda .\supset \vdash .\text{Prop} \end{array} \]

*112·302. \(\vdash .\Sigma ʻ\kappa = \Sigma ʻ(\kappa -\iota ʻ\Lambda )\)

Dem. \[ \begin{array}{l} \vdash .*112·102.\supset \vdash .\Sigma ʻ\kappa &= \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .x\in \alpha .R = x\downarrow \alpha\}\\ [*10·24] &= \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .\exists !\alpha .x\in \alpha .R = x\downarrow \alpha\}\\ [*53·52] &= \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa - \iota ʻ\Lambda .x\in \alpha .R = x\downarrow \alpha\}\\ [*112·102] &= \Sigma ʻ(\kappa - \iota ʻ\Lambda ).\supset \vdash .\text{Prop} \end{array} \]

Thus if \(\Lambda\) is a member of a class of classes, it does not affect the value of their arithmetical sum.

*112·303. \(\vdash : \kappa \cap \Lambda = \Lambda . \supset . \Sigma ʻ\kappa \cap \Sigma ʻ\lambda = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*112·102.\supset \\ \vdash :R\in \Sigma ʻ\kappa \cap \Sigma ʻ\lambda . &\equiv .(\exists \alpha ,\beta ,x,y).\alpha \in \kappa .\beta \in \lambda .x\in \alpha .y\in \beta .R = x\downarrow \alpha = y\downarrow \beta .\\ [*55·202] &\supset .(\exists \alpha ,x).\alpha \in \kappa \cap \lambda .x\in \alpha .\\ [*24·5] &\supset .\exists !\kappa \cap \lambda &\qquad \text{(1)}\\ \vdash .(1).\text{Transp}.\supset \vdash .\text{Prop} \end{array} \]

*112·304. \(\vdash :\Sigma ʻ\kappa = \Lambda . \equiv .sʻ\kappa = \Lambda\)

Dem. \[ \begin{array}{l} \vdash .*112·3·301.*53·24.&\supset \vdash :sʻ\kappa = \Lambda .\supset .\Sigma ʻ\kappa = \Lambda &\qquad \text{(1)}\\ \vdash . *112·102. &\supset \vdash :\alpha \in \kappa .x\in \alpha .\supset .x\downarrow \alpha \in \Sigma ʻ\kappa :\\ [*10·24.*40·11] &\supset \vdash :\exists !sʻ\kappa .\supset .\exists !\Sigma ʻ\kappa &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*112·31. \(\vdash .\Sigma ʻ(\kappa \cup \lambda ) = \Sigma ʻ\kappa \cup \Sigma ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*112·1.\supset \vdash .\Sigma ʻ(\kappa \cup \lambda ) &= sʻ\in \unicode{x21A7}ʻʻ(\kappa \cup \lambda )\\ [*40·31] &= sʻ\in \unicode{x21A7}ʻʻ\kappa \cup sʻ\in \unicode{x21A7}ʻʻ\lambda \\ [*112·1] &= \Sigma ʻ\kappa \cup \Sigma ʻ\lambda .\supset \vdash .\text{Prop} \end{array} \]

[Pg 103]

*112·311. \(\vdash : \kappa \cap \lambda = \Lambda . \supset . \Sigma \text{Nc}ʻ(\kappa \cup \lambda ) = \Sigma \text{Nc}ʻ\kappa +_{c} \Sigma \text{Nc}ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash . *112·303 . *110·32 . \supset \\ \vdash : \text{Hp} . \supset . \text{Nc}ʻ(\Sigma ʻ\kappa \cup \Sigma ʻ\lambda ) &= \text{Nc}ʻ\Sigma ʻ\kappa +_{c} \text{Nc}ʻ\Sigma ʻ\lambda \\ [*112·101] &= \Sigma \text{Nc}ʻ\kappa +_{c} \Sigma \text{Nc}ʻ\lambda &\qquad \text{(1)}\\ \vdash . (1) . *112·31 . \supset \vdash . \text{Prop} \end{array} \]

*112·32. \(\vdash . \Sigma ʻ\iota ʻ\alpha = \in \unicode{x21A7} \alpha\)

Dem. \[ \begin{array}{l} \vdash . *53·31 .*112·1 . \supset \vdash . \Sigma ʻ\iota ʻ\alpha &= sʻ\iota ʻ\in \unicode{x21A7} \alpha \\ [*53·02] &= \in \unicode{x21A7} \alpha . \supset \vdash . \text{Prop} \end{array} \]

*112·321. \(\vdash . \Sigma \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha \quad[*112·32·101 . *85·601]\)

*112·33. \(\vdash . \Sigma ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \in \unicode{x21A7} \alpha \cup \in \unicode{x21A7} \beta \quad[*112·32·31]\)

*112·331. \(\vdash . \Sigma ʻ(\kappa \cup \iota ʻ\beta ) = \Sigma ʻ\kappa \cup \in \unicode{x21A7} \beta \quad[*112·31·32]\)

*112·34. \(\vdash : \alpha \neq \beta . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash . *51·231 . *112·311 . \supset \\ \vdash : \text{Hp} . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) &= \Sigma \text{Nc}ʻ\iota ʻ\alpha +_{c} \Sigma \text{Nc}ʻ\iota ʻ\beta \\ [*112·321] &= \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta : \supset \vdash . \text{Prop} \end{array} \]

This proposition establishes the agreement of the two definitions of addition, namely that in *110 and that in *112. It will be seen that the definition of *112 is inapplicable to the addition of a class to itself, if this is to give the double of the class, instead of (like logical addition) simply reproducing the class. Hence the need of the condition \(\alpha\neq \beta\) in the above proposition.

*112·341. \(\vdash : \beta {\sim} \in \kappa . \supset . \Sigma \text{Nc}ʻ(\kappa \cup \iota ʻ\beta ) = \Sigma \text{Nc}ʻ\kappa +_{c} \text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash . *51·211 . \supset \vdash : \text{Hp} . &\supset . \kappa \cap \iota ʻ\beta = \Lambda .\\ [*112·311] \supset . \Sigma \text{Nc}ʻ(\kappa \cup \iota ʻ\beta ) &= \Sigma \text{Nc}ʻ\kappa +_{c} \Sigma \text{Nc}ʻ\iota ʻ\beta \\ [*112·321] &= \Sigma \text{Nc}ʻ\kappa +_{c} \text{Nc}ʻ\beta : \supset \vdash . \text{Prop} \end{array} \]

*112·35. \(\vdash : \alpha \neq \beta . \alpha \neq \gamma . \beta \neq \gamma . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma ) = \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *51·231 . *112·311 . \supset \\ \vdash : \text{Hp} . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma ) &= \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) +_{c} \Sigma \text{Nc}ʻ\iota ʻ\gamma \\ [*112·34·321] &= \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma : \supset \vdash . \text{Prop} \end{array} \]

Similar propositions can obviously be proved for any finite number of summands.

[Pg 104]

*112·4. \(\vdash :sʻ\kappa ,sʻʻ\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma \text{Nc}ʻsʻ\kappa = \Sigma \text{Nc}ʻsʻʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*112·15.\supset \vdash :\text{Hp}.\supset .\Sigma \text{Nc}ʻsʻ\kappa &= \text{Nc}ʻsʻsʻ\kappa \\ [*42·1] &= \text{Nc}ʻsʻsʻʻ\kappa \\ [*112·15] &= \Sigma \text{Nc}ʻsʻʻ\kappa :\supset \vdash .\text{Prop} \end{array} \]

*112·41. \(\vdash .sʻ\Sigma ʻʻ\lambda = \Sigma ʻsʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*112·1.\supset \vdash .sʻ\Sigma ʻʻ\lambda &= sʻsʻʻ\in \unicode{x21A7}ʻʻʻ\lambda \\ [*42·1] &= sʻsʻ\in \unicode{x21A7}ʻʻʻ\lambda \\ [*40·38] &= sʻ\in \unicode{x21A7}ʻʻsʻ\lambda\\ [*112·1] &= \Sigma ʻsʻ\lambda .\supset \vdash .\text{Prop} \end{array} \]

*112·42. \(\vdash :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\Sigma ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\)

Dem. \[ \begin{array}{l} \vdash .*112·303.\supset \vdash \colon\ldotp \lambda \in \text{Cls}^{2} \text{excl}.&\supset :\beta ,\gamma \in \lambda .\beta \neq \gamma .\supset _[\beta ,\gamma ].\Sigma ʻ\beta \cap \Sigma ʻ\gamma = \Lambda :\\ [*30·37.\text{Transp}.*37·63] &\supset :\mu ,\nu \in \Sigma ʻʻ\lambda .\mu \neq \nu .\supset _{\mu ,\nu }.\mu \cap \nu = \Lambda :\\ [*84·1] &\supset :\Sigma ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*112·43. \(\vdash :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\text{Nc}ʻ\Sigma ʻ\Sigma ʻʻ\lambda = \text{Nc}ʻ\Sigma ʻsʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*112·15·42.\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ\Sigma ʻ\Sigma ʻʻ\lambda &= \text{Nc}ʻsʻ\Sigma ʻʻ\lambda \\ [*112·41] &= \text{Nc}ʻ\Sigma ʻsʻ\lambda :\supset \vdash .\text{Prop} \end{array} \]

The above is the associative law for arithmetical addition.

Summary of *113.

In this number, we give a definition of multiplication which can be extended to any finite number of factors, but not to an infinite number of factors. We define first the arithmetical class-product of two classes \(\alpha\) and \(\beta\), and thence the product of two cardinals \(\mu\) and \(\nu\) as the number of terms in the product of \(\alpha\) and \(\beta\) when \(\alpha\) has \(\mu\) terms and \(\beta\) has \(\nu\) terms. In *114, we shall give a definition of multiplication which is not restricted to a finite number of factors. The advantages of the definition to be given in this number are, that it does not require the factors to be of the same type, and that it enables us to multiply a class by itself without (as in logical addition and multiplication) simply reproducing the class in question. The disadvantage of the definition in this number is the impossibility of extending it to an infinite number of factors.

The arithmetical class-product of two classes \(\alpha\) and \(\beta\), which we denote by \(\beta \times \alpha\)[5], is the class of all ordinal couples which take their referent from \(\alpha\) and their relatum from \(\beta\), i.e. it is the class of all such relations as \(x\downarrow y\), where \(x\in \alpha\) and \(y\in\beta\). For a given \(y\), the class of couples we obtain is \(\downarrow yʻʻ\alpha\), which is similar to \(\alpha\); and the number of such classes, for varying \(y\), is \(\text{Nc}ʻ\beta\). Thus we have \(\text{Nc}ʻ\beta\) classes of \(\text{Nc}ʻ\alpha\) couples, and \(\beta \times \alpha\) is the logical sum of these classes of couples. The class of such classes as \(\downarrow yʻʻ\alpha\), where \(y\in \beta\), is important again in connection with exponentiation; we have \(\downarrow yʻʻ\alpha =\alpha \downarrow_{,,}y\), whence the class of such classes, when \(y\) is varied among the \(\beta\)'s, is \(\alpha \downarrow_{,,}ʻʻ\beta\), and \[ \beta \times \alpha =sʻ\alpha \downarrow_{,,}ʻʻ\beta \quad(\text{cf}. *40·7), \] which we take as the definition of \(\beta \times \alpha\).

[Pg 106]

We represent the arithmetical product of \(\mu\) and \(\nu\) by \(\mu \times _{c}\nu\). This, as well as \(\text{Nc}ʻ\alpha \times_{c}\text{Nc}ʻ\beta\), is defined in terms of \(\alpha \times \beta\) exactly as, in *110, the sum was defined in terms of \(\alpha +\beta\).

The present number contains many propositions which belong to the theory of \(\alpha \downarrow_{,,}ʻʻ\beta\) rather than (specially) of \(\beta \times \alpha\); and many propositions are rather logical than arithmetical in their nature, i.e. they might have been given in *55. The line is, however, so hard to draw that it has seemed better to deal simultaneously with all propositions on \(\alpha \downarrow_{,,}ʻʻ\beta\) or on its sum, which is \(\beta\times \alpha\). Thus in the present number, the early propositions, down to *113·118, deal mainly with logical properties of \(\alpha\downarrow_{,,}ʻʻ\beta\) and \(\beta \times \alpha\); the following propositions, down to *113·13, deal mainly with arithmetical properties of \(\alpha \downarrow_{,,}ʻʻ\beta\); the propositions *113·14—·191 are concerned mainly with arithmetical properties of \(\beta \times\alpha\); *113·2—·27 deal with the simpler properties of \(\mu \times_{c}\nu\); *113·3—·34 give propositions involving the multiplicative axiom, and exhibiting the connection (assuming this axiom) of addition and multiplication; *113·4—·491 are concerned with various forms of the distributive law; *113·5—*113·541 deal with the associative law of multiplication, and the remaining propositions deal with multiplication by 0 or 1 or 2.

The most important propositions in the present number are the following:

*113·101. \(\vdash :R\in \beta \times \alpha .\equiv .(\exists x,y).x\in \alpha .y\in \beta .R=x\downarrow y\)

This merely embodies the definition of \(\beta \times \alpha\).

*113·105. \(\vdash :\exists !\alpha .\supset .\alpha \downarrow_{,,}.\in 1\rightarrow 1\)

This proposition is especially useful in dealing with exponentiation (*116).

*113·114. \(\vdash \colon\ldotp \alpha =\Lambda .\lor.\beta =\Lambda :\equiv .\beta \times \alpha =\Lambda\)

It is in virtue of this proposition that a product of a finite number of factors only vanishes when one of its factors vanishes.

*113·118. \(\vdash .sʻ\text{D}ʻʻ(\beta \times \alpha )\subset \alpha .sʻ\text{ᗡ}ʻʻ(\beta \times \alpha )\subset \beta\)

This proposition is chiefly useful in the analogous theory of ordinal products (*165, *166), where it enables us to apply *74·773. Unless \(\beta =\Lambda\), we have \(sʻ\text{D}ʻʻ(\beta \times \alpha)=\alpha\), and unless \(\alpha =\Lambda ,sʻ\text{ᗡ}ʻʻ(\beta \times \alpha )=\beta\) (*113·116).

*113·12. \(\vdash :\exists !\alpha .\supset .\alpha \downarrow_{,,}ʻʻ\beta \in \text{Nc}ʻ\beta \cap \text{Cl excl}ʻ\text{Nc}ʻ\alpha\)

I.e. unless \(\alpha\) is null, \(\alpha\downarrow_{,,}ʻʻ\beta\) consists of \(\text{Nc}ʻ\beta\) mutually exclusive classes each having \(\text{Nc}ʻ\alpha\) members.

*113·127. \[\begin{aligned}\vdash :R\upharpoonright y\in \alpha \,\overline{\text{ sm }}\, \gamma .S\upharpoonright \delta \in \beta &\,\overline{\text{ sm }}\, \delta .\supset .\\ &(R\parallel \breve{S} )\upharpoonright (\delta \times \gamma )\in (\alpha \downarrow_{,,}ʻʻ\beta )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,}ʻʻ\delta )\end{aligned}\]

[Pg 107]

This is an important proposition, since it gives a double correlator of \(\alpha \downarrow_{,,}ʻʻ\beta\) with \(\gamma\downarrow_{,,}ʻʻ\delta\) whenever simple correlators of \(\alpha\) with \(\gamma\) and of \(\beta\) with \(\delta\) are given. It leads at once to

*113·13. \(\vdash : \alpha \text{ sm } \gamma . \beta \text{ sm } \delta . \supset . \alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \text{ sm } \gamma \downarrow_{,,} ʻʻ\delta . (\beta \times \alpha ) \text{ sm } (\delta \times \gamma )\)

This proposition is fundamental in the theory of multiplication, since it shows that the number of members of \(\beta \times \alpha\) depends only upon the numbers of members of \(\alpha\) and \(\beta\). It is also fundamental in the theory of exponentiation, as will appear in *116.

*113·141. \(\vdash . \text{Nc}ʻ(\alpha \times \beta ) = \text{Nc}ʻ(\beta \times \alpha )\)

This is the source of the commutative law of multiplication (*113·27).

*113·146. \(\vdash : \alpha \neq \beta . \supset . \alpha \times \beta \text{ sm } {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\)

This connects our present theory of multiplication with the theory of selections.

We come next to propositions concerning \(\mu \times _{c} \nu\). We have

*113·204. \(\vdash \colon\ldotp \mu = \Lambda . \lor . \nu = \Lambda . \lor . {\sim} (\mu , \nu \in \text{NC}) : \supset . \mu \times _{c} \nu = \Lambda\)

The use of this proposition, like that of *110·4, is for avoiding trivial exceptions.

*113·23. \(\vdash . \mu \times _{c} \nu \in \text{NC}\)

*113·25. \(\vdash . \text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta = \text{Nc}ʻ(\gamma \times \delta )\)

This proposition enables us to infer propositions on products of cardinals from propositions on products of classes, and is therefore constantly used.

*113·27. \(\vdash . \mu \times _{c} \nu = \nu \times _{c} \mu\)

This is the commutative law of cardinal multiplication.

The chief proposition using the multiplicative axiom is

*113·31. \(\vdash \colon\ldotp \text{Mult ax} . \supset : \mu , \nu \in \text{NC} . \kappa \in \nu \cap \text{Cl}ʻ\mu . \supset . \Sigma ʻ\kappa \in \mu \times _{c} \nu\)

I.e. assuming the multiplicative axiom, the sum of the numbers of members in \(\nu\) classes of \(\mu\) terms is \(\mu \times _{c}\nu\). If we had taken this sum as defining \(\mu \times _{c}\nu\), almost all propositions on multiplication would have required the multiplicative axiom. The advantage of \(\alpha \downarrow_{,,}ʻʻ\beta\) is that, given \(\alpha \text{ sm } \gamma\) and \(\beta\text{ sm } \delta\), we can construct a double correlator of \(\alpha\downarrow_{,,} ʻʻ\beta\) with \(\gamma \downarrow_{,,} ʻʻ\delta \), without using the multiplicative axiom. This is proved in *113·127 (mentioned above).

The distributive law, which is next considered, has various forms. We have, to begin with,

*113·4. \(\vdash . (\beta \cup \gamma ) \times \alpha = (\beta \times \alpha ) \cup (\gamma \times \alpha )\)

whence, using also the commutative law, we easily deduce

*113·43. \(\vdash . (\nu +_{c} \varpi ) \times _{c} \mu = \mu \times _{c} (\nu +_{c} \varpi ) = (\mu \times _{c} \nu ) +_{c} (\mu \times _{c} \varpi )\)

[Pg 108]

But the distributive law also holds when, instead of enumerated summands \(\beta\), \(\gamma\) or \(\nu\), \(\varpi\), the summands are given as the members of a class \(\kappa\), which may be infinite. We have

*113·48. \(\vdash . sʻ\alpha \times ʻʻ\kappa = \alpha \times sʻ\kappa = \text{Cnv}ʻʻ(sʻ\kappa \times \alpha )\)

whence, using the definitions of *112, we find

*113·491. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \Sigma \text{Nc}ʻ\alpha \times ʻʻ\kappa = \text{Nc}ʻ(\alpha \times \Sigma ʻ\kappa ) = \text{Nc}ʻ\alpha \times _{c} \Sigma \text{Nc}ʻ\kappa\)

This is an extension of the distributive law to the case where the number of summands may be infinite.

The associative law

*113·54. \(\vdash . (\mu \times _{c} \nu ) x_{c} \varpi = \mu \times _{c} (\nu \times _{c} \varpi )\)

is proved without any difficulty.

We prove next that \(\mu \times _{c} \nu = 0\) when, and only when, \(\mu = 0\) or \(\nu = 0\), \(\mu\), \(\nu\) being existent cardinals (*113·602); that a cardinal is unchanged when it is multiplied by 1 (*113·62·621); that \(\mu \times _{c} 2 = \mu +_{c} \mu\) (*113·66) and that \(\mu \times _{c} (\nu + 1) = (\mu \times _{c} \nu ) +_{c} \mu\) (*113·671).

*113·02. \(\beta \times \alpha = sʻ\alpha \downarrow_{,,} ʻʻ\beta \quad\text{Df}\)

*113·03. \(\mu \times _{c} \nu = \hat{\xi} \{(\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu \text{N}_{0}\text{c}ʻ\beta . \xi \text{ sm } (\alpha \times \beta )\} \quad\text{Df}\)

*113·04. \(\text{Nc}ʻ\beta \times _{c} \mu = \text{N}_{0}\text{c}ʻ\beta \times _{c} \mu \quad\text{Df}\)

*113·05. \(\mu \times _{c} \text{Nc}ʻ\alpha = \mu \times _{c} \text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)

In relation to types, *113·03·04·05 call for similar remarks to those made in *110 for addition.

*113·1. \(\vdash . \beta \times \alpha = sʻ\alpha \downarrow_{,,} ʻʻ\beta \quad[(*113·02)]\)

*113·101. \(\vdash : R \in \beta \times \alpha . \equiv . (\exists x, y) . x \in \alpha . y \in \beta . R = x\downarrow y \quad[*40·7 . *113·1]\)

*113·102. \(\vdash : y \in \beta . \supset . \alpha \downarrow_{,,} y = (\alpha \uparrow \beta )_{\Delta }ʻ\iota ʻy\)

Dem. \[ \begin{array}{l} \vdash . *35·103 . \supset \\ \vdash \colon\ldotp \text{Hp} . \supset : x (\alpha \uparrow \beta ) y . \equiv . x \in \alpha :\\ [*85·51] \supset : (\alpha \uparrow \beta )_{\Delta }ʻ\iota ʻy = \downarrow yʻʻ\alpha \\ [(*38·03)] = \alpha \downarrow_{,,} y : \supset \vdash . \text{Prop} \end{array} \]

*113·103. \(\vdash . \alpha \downarrow_{,,} ʻʻ\beta = (\alpha \uparrow \beta )_{\Delta }ʻʻ\iotaʻʻ\beta = (\alpha \uparrow \beta ) \unicode{x21A7} ʻʻ\beta \quad[*113·102 . *85·52]\)

[Pg 109]

*113·104. \(\vdash . \text{E}! \alpha \downarrow_{,,} ʻy \quad[*38·12]\)

*113·105. \(\vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,} \in 1 \rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash . *113·104. *71·166 . &\supset \vdash . \alpha \downarrow_{,,} \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash . *38·131. \supset \vdash : \alpha \downarrow_{,,} ʻy = \alpha \downarrow_{,,} ʻz . x \in \alpha . &\supset . x \downarrow y \in \alpha \downarrow_{,,} ʻz .\\ [*38·131] &\supset . (\exists xʻ) . xʻ \in \alpha . x \downarrow y = xʻ \downarrow z.\\ [*55·202] &\supset . y = z &\qquad \text{(2)}\\ \vdash .(2). *10·11·23·35. &\supset \vdash : \exists ! \alpha .\alpha \downarrow_{,,} ʻy = \alpha \downarrow_{,,} ʻz . \supset . y = z &\qquad \text{(3)}\\ \vdash .(1).(3). *71·54. \supset \vdash . \text{Prop} \end{array} \]

*113·106. \(\vdash : x \in \alpha . y \in \beta . \supset . x \downarrow y \in \beta \times \alpha \quad[*113·101]\)

*113·107. \(\vdash : \exists ! \alpha . \exists ! \beta . \supset . \exists ! \beta \times \alpha \quad [*113·106]\)

*113·11. \(\vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,}ʻʻ\beta \in \text{Nc}ʻ\beta : (y) . \alpha \downarrow_{,,} y \in \text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash . *113·105·104 . *73·26 . &\supset \vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \beta &\qquad \text{(1)}\\ \vdash . *38·2. *73·611. & \supset \vdash . \alpha \downarrow_{,,} y \text{ sm } \alpha &\qquad \text{(2)}\\ \vdash . (1) . (2) . &\supset \vdash . \text{Prop} \end{array} \]

*113·111. \(\vdash . \alpha \downarrow_{,,} ʻʻ\beta \in \text{Cls}^{2} \text{excl} \quad[*113·103 . *85·55]\)

*113·112. \(\vdash : \alpha = \Lambda . \exists ! \beta . \supset . \alpha \downarrow_{,,} ʻʻ\beta = \iota ʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash . *38·3 . \supset \vdash : \text{Hp} . \supset . \alpha \downarrow_{,,} ʻʻ\beta &= \hat{\mu} \{ (\exists y).y \in \beta . \mu = \downarrow yʻʻ\Lambda \}\\ [*37·29] &= \hat{\mu} \{ (\exists y).y \in \beta . \mu = \Lambda\}\\ [\text{Hp}] & = \iota ʻ\Lambda \end{array} \]

*113·113. \(\vdash : \beta = \Lambda . \supset . \alpha \downarrow_{,,} ʻʻ\beta = \Lambda \quad[*37·29]\)

*113·114. \(\vdash \colon\ldotp \alpha = \Lambda .\lor.\beta = \Lambda : \equiv .\beta \times \alpha = \Lambda \quad[*113·1·112·113·107.*53·24]\)

*113·115. \(\vdash . \dot{s} ʻ(\beta \times \alpha ) = \alpha \uparrow \beta\)

Dem. \[ \begin{array}{l} \vdash . *113·101.*41·11. \supset \\ \vdash : u \{ \dot{s} ʻ(\beta \times \alpha )\} v. &\equiv . (\exists R, x, y) . x \in \alpha . y \in \beta . R = x \downarrow y . uRv .\\ [*13·195.*55·13] &\equiv . (\exists x, y) . x \in \alpha . y \in \beta . u = x . v = y .\\ [*13·22] &\equiv . u \in \alpha . v \in \beta .\\ [*35·103] &\equiv . u (\alpha \uparrow \beta ) v : \supset \vdash . \text{Prop} \end{array} \]

*113·116. \[\begin{aligned}&\vdash : \exists ! \beta . \supset . sʻ\text{D}ʻʻ(\beta \times \alpha ) = \alpha : \exists ! \alpha . \supset . sʻ\text{ᗡ}ʻʻ(\beta \times \alpha ) = \beta \\ &[*113·115 . *41·43·44. *35·85·86]\end{aligned}\]

*113·117. \[\begin{aligned}&\vdash \colon\ldotp \alpha = \Lambda . \lor . \beta = \Lambda : \supset . sʻ\text{D}ʻʻ(\beta \times \alpha ) = \Lambda . sʻ\text{ᗡ}ʻʻ(\beta \times \alpha ) = \Lambda\\ &[*113·115 . *41·43·44. *35·88]\end{aligned}\]

*113·118. \(\vdash . sʻ\text{D}ʻʻ(\beta \times \alpha ) \subset \alpha . sʻ\text{ᗡ}ʻʻ(\beta \times \alpha ) \subset \beta \quad[*113·116·117]\)

[Pg 110]

*113·12. \(\vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,} ʻʻ\beta \in \text{Nc}ʻ\beta \cap \text{Cl excl}ʻ\text{Nc}ʻ\alpha \quad[*113·11·111]\)

*113·121. \(\vdash . \Sigma ʻ\alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \beta \times \alpha \quad[*112·15 . *113·111·1]\)

*113·122. \[\begin{aligned}&\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in \text{Cls} \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . \supset . (R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in 1 \rightarrow 1\\ &[*74·773. *113·118]\end{aligned}\]

*113·123. \[\begin{aligned}\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in 1 \rightarrow \text{Cls}. \gamma &\subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . z \in \gamma . w \in \delta . \supset .\\ &(R \parallel \breve{S} )ʻ(z \downarrow w) = (Rʻz) \downarrow (Sʻw) \quad[*55·61]\end{aligned}\]

*113·124. \[\begin{aligned}\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in 1 \rightarrow \text{Cls}. \gamma &\subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . w \in \delta . \supset .\\ &(R \parallel \breve{S} )ʻʻ\gamma \downarrow_{,,} w = (Rʻʻ\gamma ) \downarrow_{,,} (Sʻw)\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *113·123 . *38·131 . \supset \vdash : \text{Hp} . &\supset . (R \parallel \breve{S} )ʻʻ \downarrow wʻʻ\gamma = \downarrow (Sʻw)ʻʻRʻʻ\gamma .\\ [*38·2] &\supset . (R \parallel \breve{S} )ʻʻ\gamma \downarrow_{,,} w = (Rʻʻ\gamma ) \downarrow_{,,} (Sʻw) : \supset \vdash . \text{Prop} \end{array} \]

*113·125. \[\begin{aligned}\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in 1 \rightarrow &\text{Cls}. \gamma \subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . \supset .\\ &(R \parallel \breve{S} )_{\in }ʻʻ\gamma \downarrow_{,,} ʻʻ\delta = (Rʻʻ\gamma ) \downarrow_{,,} ʻʻ(Sʻʻ\delta ) \quad[*113·124]\end{aligned}\]

*113·126. \(\vdash : \text{Hp} *113·125 . \supset . (R \parallel \breve{S} )ʻʻ(\delta \times \gamma ) = (Sʻʻ\delta ) \times (Rʻʻ\gamma )\)

Dem. \[ \begin{array}{l} \vdash . *113·1 . *40·38 . &\supset \vdash . (R \parallel \breve{S} )ʻʻ(\delta \times \gamma ) = sʻ(R \parallel \breve{S} )ʻʻʻ\gamma \downarrow_{,,} ʻʻ\delta &\qquad \text{(1)}\\ \vdash . (1) . *113·125 . \supset \vdash : \text{Hp} . \supset . (R \parallel \breve{S} )ʻʻ(\delta \times \gamma ) &= sʻ(Rʻʻ\gamma ) \downarrow_{,,} ʻʻ(Sʻʻ\delta )\\ [*113·1] &= (Sʻʻ\delta ) \times (Rʻʻ\gamma ) : \supset \vdash . \text{Prop} \end{array} \]

*113·127. \[\begin{aligned}\vdash : R \upharpoonright \gamma \in \alpha &\,\overline{\text{ sm }}\, \gamma . S \upharpoonright \delta \in \beta \,\overline{\text{ sm }}\, \delta . \supset .\\ &(R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\alpha \downarrow_{,,} ʻʻ\beta ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,} ʻʻ\delta )\\ &[*113·122·125 . *43·302 . *73·142 . *111·14]\end{aligned}\]

*113·128. \[\begin{aligned}\vdash : \text{Hp} &*113·127 . \supset . (R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\beta \times \alpha ) \,\overline{\text{ sm }}\, (\delta \times \gamma ).\\ &(R \parallel \breve{S} )_{\in } \upharpoonright (\gamma \downarrow_{,,} ʻʻ\delta ) \in (\alpha \downarrow_{,,} ʻʻ\beta ) \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,} ʻʻ\delta ) \quad[*113·127 . *111·15]\end{aligned}\]

*113·13. \[\begin{aligned}&\vdash : \alpha \text{ sm } \gamma . \beta \text{ sm } \delta . \supset . \alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \text{ sm } \gamma \downarrow_{,,} ʻʻ\delta . (\beta \times \alpha ) \text{ sm } (\delta \times \gamma )\\ &[*113·127 . *111·4·44 . *113·1]\end{aligned}\]

*113·14. \(\vdash . \alpha \times \beta = \text{Cnv}ʻʻ(\beta \times \alpha ) \quad[*113·101 . *55·14]\)

*113·141. \(\vdash . \text{Nc}ʻ(\alpha \times \beta ) = \text{Nc}ʻ(\beta \times \alpha ) \quad[*113·14 . *73·4]\)

*113·142. \(\vdash : \exists ! \beta . \supset . \text{D}ʻʻ(\beta \times \alpha ) = \iotaʻʻ\alpha : \exists ! \alpha . \supset . \text{ᗡ}ʻʻ(\beta \times \alpha ) = \iotaʻʻ\beta \)

[Pg 111]

Dem. \[ \begin{array}{l} \vdash . *55·261 . *2·02 . &\supset \vdash : y \in \beta .\supset . \text{D}ʻʻ\alpha \downarrow_{,,} y = \iotaʻʻ\alpha \\ [*37·63] &\supset \vdash : \gamma \in \text{D}ʻʻʻ \alpha \downarrow_{,,} ʻʻ\beta . \supset . \gamma = \iotaʻʻ\alpha &\qquad \text{(1)}\\ \vdash . *37·45. \supset \vdash :\exists !\beta .&\supset .\exists !\text{D}ʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).*51·141. \supset \vdash :\exists !\beta .&\supset .\text{D}ʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta = \iota ʻ\iotaʻʻ\alpha .\\ [*40·38.*53·02] &\supset .\text{D}ʻʻsʻ\alpha \downarrow_{,,}ʻʻ\beta = \iotaʻʻ\alpha &\qquad \text{(3)}\\ \vdash . *55·251. \supset \vdash :\exists !\alpha .&\supset .\text{ᗡ}ʻʻ\alpha \downarrow_{,,}y = \iota ʻ\iota ʻy.\\ [*37·355] &\supset .\text{ᗡ}ʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta = \iotaʻʻ\iotaʻʻ\beta .\\ [*40·38.*53·22] &\supset .\text{ᗡ}ʻʻsʻ\alpha \downarrow_{,,}ʻʻ\beta = \iotaʻʻ\beta &\qquad \text{(4)}\\ \vdash .(3).(4).*113·1. \supset \vdash .\text{Prop} \end{array} \]

*113·143. \[\begin{aligned}\vdash :\alpha \neq \beta .P = x\downarrow y.&R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta .\supset .\\ &P = (Rʻ\alpha )\downarrow (Rʻ\beta ).R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*55·62.\supset \vdash :\text{Hp}.&\supset .Rʻ\alpha = x.Rʻ\beta = y .\\ [*30·19.*13·15] &\supset .P = (Rʻ\alpha )\downarrow (Rʻ\beta ) &\qquad \text{(1)}\\ \vdash .*55·15.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻP = \iota ʻx.\text{ᗡ}ʻP = \iota ʻy.\\ [*55·1] &\supset .R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*113·144. \[\begin{aligned}\vdash :\alpha &\neq \beta .T = \hat{P} \hat{R} \{(\exists x,y).x\in \alpha .y\in \beta .P = x\downarrow y.R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta\}.\\ &\supset .T\in 1\rightarrow 1.\text{D}ʻT = \beta \times \alpha .\text{ᗡ}ʻT = {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ &PTR.QTR.\supset .(\exists x,y,z,w).x,z\in \alpha .y,w\in \beta .P = x\downarrow y.Q = z\downarrow w.\\ &R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta = z\downarrow \alpha \unicode{x228d} w\downarrow \beta .\\ [*113·143]&\supset .P = (Rʻ\alpha )\downarrow (Rʻ\beta ).Q = (Rʻ\alpha )\downarrow (Rʻ\beta ).\\ [*13·172] &\supset .P = Q &\qquad \text{(1)}\\ \vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :PTQ.PTR.\supset .\\ &(\exists x,y,z,w).x,z\in \alpha .y,w\in \beta .P = x\downarrow y = w\downarrow z.Q = x\downarrow \alpha \unicode{x228d} y\downarrow \beta .R=z\downarrow \alpha \unicode{x228d} w\downarrow \beta .\\ [*113·143]&\supset .Q = DʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta .R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta .\\ [*13·172] &\supset .Q = R &\qquad \text{(2)}\\ \vdash .*33·13. &\supset \vdash :\text{Hp}.\supset .\\ &\text{D}ʻT = \hat{P} \{(\exists R,x,y).x\in \alpha .y\in \beta .P = x\downarrow y.R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta\}\\ [*11·55.*13·19] &= \hat{P} \{(\exists x,y).x\in \alpha .y\in \beta .P = x\downarrow y\}\\ [*113·101] &= \beta \times \alpha &\qquad \text{(3)}\\ \vdash .*33·131.\supset \vdash :\text{Hp}.\supset .\\ \text{ᗡ}ʻT &= \hat{R} \{(\exists P,x,y).x\in \alpha .y\in \beta .P = x\downarrow y.R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta\}\\ [*11·55.*13·19] &= \hat{R} \{(\exists x,y).x\in \alpha .y\in \beta .R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta \}\\ [*80·9] &= {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) &\qquad \text{(4)}\\ \vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} \end{array} \]

[Pg 112]

Note to *113·144. In virtue of *113·143 and *55·61 we have \[ \vdash \colon\ldotp \text{Hp} *113·144.\supset :PTR. \equiv .R\in {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ).P = (R\parallel \breve{R} )ʻ(\alpha \downarrow \beta ). \]

At a later stage (in *150) we shall put \[ R\dagger S = (R\parallel \breve{R} )ʻS \quad\text{Df}. \]

Thus we shall have, anticipating this notation, \[ \vdash : \text{Hp} *113·144.\supset .T = \{\dagger(\alpha \downarrow \beta )\}\upharpoonright {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ). \]

Hence we have \[ \vdash :\alpha \neq \beta .\supset .\{\dagger(\alpha \downarrow \beta )\}\upharpoonright {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\in (\beta \times \alpha )\,\overline{\text{ sm }}\, {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ). \]

*113·145. \(\vdash :\alpha \neq \beta .\supset .\beta \times \alpha \text{ sm }{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \quad[*113·144]\)

*113·146. \(\vdash :\alpha \neq \beta .\supset .\alpha \times \beta \text{ sm } {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \quad[*113·141·145]\)

*113·147. \[\begin{aligned}\vdash :\text{Hp}*133·144.&\beta \times \alpha = \mu .\supset .\\ &T = \hat{P} \hat{R} \{P\in \mu .R = \text{D}ʻP\uparrow \iota ʻsʻ\text{D}ʻʻ\mu \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻsʻ\text{ᗡ}ʻʻ\mu \}\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*113·114.\text{Transp}.\supset \vdash :\text{Hp}.P\in \mu .&\supset .\exists !\alpha .\exists !\beta .\\ [*113·142.*53·22]&\supset .\alpha = sʻ\text{D}ʻʻ\mu .\beta = sʻ\text{ᗡ}ʻʻ\mu &\qquad \text{(1)}\\ \vdash .*133·101·143.&\supset \vdash \colon\ldotp \text{Hp}.P\in \mu .\supset :PTR. \equiv .R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta &\qquad \text{(2)}\\ \vdash .*113·144.&\supset \vdash :\text{Hp}.PTR.\supset .P\in \mu &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*113·101.\supset \vdash .\text{Prop} \end{array} \]

The advantage of this proposition is that it exhibits the correlator of \(\beta \times \alpha\) and \(({\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iotaʻ\beta)\) as a function of \(\beta \times \alpha\).

*113·148. \(\vdash :\alpha \cap \beta = \Lambda .\supset .C\upharpoonright (\alpha \times \beta )\in 1\rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash .*113·101.*55·15.\supset \\ &\vdash \colon\ldotp \text{Hp}.\supset :R,S\in \alpha \times \beta .CʻR = CʻS.\equiv .\\ &(\exists x,x',y,y').x,x'\in \alpha .y,y'\in \beta .R = y\downarrow x,S = y'\downarrow x'.\iota ʻx\cup \iota ʻy = \iota ʻx'\cup \iota ʻy'.\\ [*54·6] &\supset .(\exists x,x',y,y').x,x'\in \alpha .y,y'\in \beta .R = y\downarrow x.S = y'\downarrow x'.x = x'.y = y'.\\ [*13·22·172] &\supset .R = S &\qquad \text{(1)}\\ \vdash .(1).*71·55.\supset \vdash .\text{Prop} \end{array} \]

*113·15. \(\vdash .Cʻʻ(\alpha \times \beta ) = Cʻʻ(\beta \times a) = \hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\)

Dem. \[ \begin{array}{l} \vdash .*113·1.*40·38.\supset \vdash .Cʻʻ(\beta \times \alpha ) &= sʻCʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta \\ [*40·4] &= \hat{\xi} \{(\exists y).y\in \beta .\xi \in Cʻʻ\alpha \downarrow_{,,}y\}\\ [*55·27.*38·2] &= \hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\} &\qquad \text{(1)}\\ \vdash .(1)\frac{\beta,\,\alpha}{\alpha,\,\beta}. \supset \vdash .Cʻʻ(\alpha \times \beta ) &= \hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 113]

*113·151. \(\vdash :\alpha \neq \beta .\supset .Cʻʻ(\alpha \times \beta ) = \text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \quad[*113·15.*80·92]\)

*113·152. \(\vdash :\alpha \cap \beta = \Lambda .\supset .Cʻʻ(\alpha \times \beta )\text{ sm }(\alpha \times \beta ).\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\text{ sm }(\alpha \times \beta )\)

Dem. \[ \begin{array}{l} \vdash .*84·41·62.&\supset \vdash :\text{Hp}.\alpha \neq \beta .\supset .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\text{ sm }{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) &\qquad \text{(1)}\\ \vdash .(1).*113·146·151.\supset \\ \vdash :\text{Hp}.\alpha \neq \beta .&\supset .Cʻʻ(\alpha \times \beta )\text{ sm }(\alpha \times \beta ).\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\text{ sm }\alpha \times \beta &\qquad \text{(2)}\\ \vdash .*24·38.\supset \vdash :\text{Hp}.\alpha = \beta .&\supset .\alpha = \Lambda .\beta = \Lambda .\\ [*113·114.*83·11.*37·29] &\supset .\alpha \times \beta = \Lambda .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \Lambda .Cʻʻ(\alpha \times \beta )=\Lambda .\\ [*73·47] &\supset .Cʻʻ(\alpha \times \beta )\text{ sm }(\alpha \times \beta ).\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\text{ sm }(\alpha \times \beta ) &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

The following proposition is only significant when \(\lambda\) and \(\mu\) are classes of relations. It is used in relation-arithmetic (*172·34).

*113·153. \(\vdash :\dot{s} ʻ\lambda \dot{\cap} \dot{s} ʻ\mu = \dot{\Lambda} .\supset .\dot{s} \mid C\upharpoonright (\lambda \times \mu )\in (sʻ\lambda \unicode{x228d}_{,,}ʻʻ\mu )\,\overline{\text{ sm }}\, (\lambda \times \mu ).sʻ\lambda \unicode{x228d}_{,,}ʻʻ\mu \text{ sm }\lambda \times \mu\)

Dem. \[ \begin{array}{l} \vdash .*55·15.*53·13.&\supset \vdash :R = T\downarrow S.\supset .\dot{s} ʻCʻR = S\unicode{x228d} T &\qquad \text{(1)}\\ \vdash .(1).*113·101.\supset \\ &\vdash :R,R'\in \lambda \times \mu .\dot{s} ʻCʻR = \dot{s} ʻCʻR'.\supset.\\ &(\exists S,S',T,T').S,S'\in \lambda .T,T'\in \mu .R = T\downarrow S.R'=T'\downarrow S'.S\unicode{x228d} T=S'\unicode{x228d} T' &\qquad \text{(2)}\\ \vdash .(2).*25·48.*41·13.\supset \\ &\vdash \colon\ldotp \text{Hp}.\supset :R,R'\in \lambda \times \mu .\dot{s} ʻCʻR = \dot{s} ʻCʻR'.\supset .R = R' &\qquad \text{(3)}\\ \vdash .(1).*113·101.&\supset \vdash .\dot{s} ʻʻCʻʻ(\lambda \times \mu ) = \hat{M} \{(\exists S,T).S\in \lambda .T\in \mu .M = S\unicode{x228d} T\}\\ [*40·7] & =sʻ\lambda \unicode{x228d}_{,,}ʻʻ\mu &\qquad \text{(4)}\\ \vdash .(3).(4).*73·25.\supset \vdash .\text{Prop} \end{array} \]

*113·16. \[\begin{aligned}\vdash :&tʻ\alpha = tʻ\beta .\supset .\text{Nc}ʻ(\alpha \times \beta ) =\\ &\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda .\xi \text{ sm }\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\}\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*113·152.\supset \vdash \colon\ldotp \gamma \in \text{N}^{1}\text{c}ʻ\alpha .&\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda .\supset :\\ \xi \text{ sm }\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta ). &\equiv .\xi \text{ sm }(\gamma \times \delta ).\\ [*113·13.*104·101] &\equiv .\xi \text{ sm }(\alpha \times \beta ).\\ [*100·31] &\equiv .\xi \in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ \vdash .(1).*5·32.*11·11·341.\supset \\ \vdash \colon\ldotp (\exists \gamma ,\delta ).\gamma \in &\text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda .\xi \text{ sm }\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta ). \equiv :\\ &(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda .\xi \in \text{Nc}ʻ(\alpha \times \beta ):\\ [*11·45] &\equiv :(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda :\xi \in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ \vdash .(2). *104·43.\supset \vdash .\text{Prop} \end{array} \]

[Pg 114]

*113·17. \(\vdash .\beta \times \alpha \in tʻtʻ(\alpha \uparrow \beta)\)

Dem. \[ \begin{array}{l} \vdash . *113·115.*41·13.\supset \vdash :R\in \beta \times \alpha .&\supset .R\,\unicode{x2abd}\, \alpha \uparrow \beta .\\ [*64·201] &\supset .R\in tʻ(\alpha \uparrow \beta ) &\qquad \text{(1)}\\ \vdash .(1).*63·5.\supset \vdash .\text{Prop} \end{array} \]

*113·171. \(\vdash :\alpha \cap \beta = \Lambda .\supset .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha \times \beta )\)

Dem. \[ \begin{array}{l} \vdash .*113·152·15.&\supset \vdash :\text{Hp}.\supset .\hat{\xi}\{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ \vdash .*51·16. & \supset \vdash :x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy.\supset .x\in \alpha .x\in \xi .\\ [*63·13] &\supset .\xi \in tʻ\alpha &\qquad \text{(2)}\\ \vdash .(2).*11·11·35.\supset \\ &\vdash .\hat{\xi}\{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\subset tʻ\alpha .\\ [*63·5] &\supset \vdash .\hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\in tʻtʻ\alpha &\qquad \text{(3)}\\ \vdash .(1).(3).&\supset \vdash :\text{Hp}.\supset .\exists !\text{Nc}ʻ(\alpha \times \beta )\cap tʻtʻ\alpha &\qquad \text{(4)}\\ \vdash .(4).*102·6.\supset \vdash .\text{Prop} \end{array} \]

Note that the hypothesis \(\alpha \cap \beta = \Lambda\) is only significant when \(\alpha\) and \(\beta\) are of the same type.

*113·172. \(\vdash :\alpha \in tʻ\beta .\supset .\exists !\text{Nc}(t^{2}ʻ\alpha )ʻ(\alpha \times \beta )\)

Dem. \[ \begin{array}{l} \vdash .*113·16. \supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \in \text{N}^{1}\text{c}ʻ\alpha .&\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda .\supset .\\ &\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ \vdash .(1).*104·43.&\supset \vdash :\text{Hp}.\supset .\\ &(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ \vdash .*104·1.\supset \vdash :\gamma \in \text{N}^{1}\text{c}ʻ\alpha .&\supset .\gamma \in t^{2}ʻ\alpha .\\ [*63·61·621] &\supset .\iota ʻ\gamma \cup \iota ʻ\delta \in tʻt^{2}ʻ\alpha .\\ [*83·81] &\supset .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\in tʻt^{2}ʻ\alpha &\qquad \text{(3)}\\ \vdash .(2).(3).&\supset \vdash :\text{Hp}.\supset .\exists !\text{Nc}ʻ(\alpha \times \beta )\cap tʻt^{2}ʻ\alpha &\qquad \text{(4)}\\ \vdash .(4).*102·6.\supset \vdash .\text{Prop} \end{array} \]

*113·18. \(\vdash :\exists !\alpha .\exists !\beta .\alpha \times \beta = \alpha' \times \beta'.\supset .\alpha = \alpha'.\beta = \beta'\)

Dem. \[ \begin{array}{l} \vdash .*113·114.\supset \vdash :\text{Hp}.&\supset .\exists !\alpha' \times \beta'.\\ [*113·114] &\supset .\exists !\alpha'.\exists !\beta' &\qquad \text{(1)}\\ \vdash .*30·37. \supset \vdash :\text{Hp}.&\supset .sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) = sʻ\text{ᗡ}ʻʻ(\alpha' \times \beta').\\ [*113·142.(1)] & \supset .sʻtʻʻ\alpha = sʻ\iota ʻʻ\alpha'.\\ [*53·22] &\supset .\alpha = \alpha' &\qquad \text{(2)}\\ \text{Similarly}\quad \vdash :\text{Hp}.&\supset .\beta = \beta' &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

[Pg 115]

*113·181. \(\vdash :\exists !\alpha .\exists !\alpha'.\alpha \times \beta = \alpha' \times \beta'.\supset .\beta = \beta'\)

Dem. \[ \begin{array}{l} \vdash . *13·172.&\supset \vdash :\beta = \Lambda .\beta' = \Lambda .\supset .\beta = \beta' &\qquad \text{(1)}\\ \vdash . *113·18.&\supset \vdash :\text{Hp}.{\sim}(\beta = \Lambda .\beta' = \Lambda ).\supset .\beta = \beta' &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash .\text{Prop} \end{array} \]

*113·182. \[\begin{aligned}\vdash :&\exists !\beta .\exists !\beta'.\alpha \times \beta = \alpha' \times \beta'.\supset .\alpha = \alpha'\\ &[\text{Proof as in *113·181}]\end{aligned}\]

*113·183. \(\vdash :\exists !\alpha .\exists !\beta .\supset .Fʻʻ(\alpha \times \beta ) = sʻCʻʻ(\alpha \times \beta ) = \alpha \cup \beta\)

Dem. \[ \begin{array}{l} \vdash .*40·57. \supset \vdash .sʻCʻʻ(\alpha \times \beta ) = sʻ\text{D}ʻʻ(\alpha \times \beta )\cup sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ \vdash .*40·56. \supset \vdash .Fʻʻ(\alpha \times \beta ) = sʻCʻʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ \vdash . *113·142. \supset \vdash :\text{Hp}.\supset .sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) = sʻ\iotaʻʻ\alpha\\ [*53·22] = \alpha &\qquad \text{(3)}\\ \vdash . *113·142. \supset \vdash :\text{Hp}.\supset .sʻ\text{D}ʻʻ(\alpha \times \beta ) = sʻ\iotaʻʻ\beta \\ [*53·22] = \beta &\qquad \text{(4)}\\ \vdash .(3).(4). \supset \vdash :\text{Hp}.\supset .sʻ\text{D}ʻʻ(\alpha \times \beta )\cup sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) = \alpha \cup \beta &\qquad \text{(5)}\\ \vdash .(1).(2).(5).\supset \vdash .\text{Prop} \end{array} \]

*113·19. \(\vdash :\exists !(\alpha \times \beta )\cap (\gamma \times \delta ). \equiv .\exists !\alpha \cap \gamma .\exists !\beta \cap \delta\)

Dem. \[ \begin{array}{l} \vdash . *113·101.&\supset \vdash \colon\ldotp \exists !(\alpha \times \beta )\cap (\gamma \times \delta ). \equiv :\\ &(\exists x,y,z,w).x\in \alpha .y\in \beta .z\in \gamma .w\in \delta .x\downarrow y = w\downarrow z:\\ [*55·202]&\equiv :(\exists x,y,z,w).x\in \alpha .y\in \beta .z\in \gamma .w\in \delta .x = z.y = w:\\ [*13·22] &\equiv :(\exists x,y).x\in \alpha \cap \gamma .y\in \beta \cap \delta \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*113·191. \(\vdash \colon\ldotp \exists !\alpha .\supset :\exists !\alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma . \equiv .\exists !\beta \cap \gamma\)

Dem. \[ \begin{array}{l} \vdash . *37·6.&\supset \vdash :\exists !\alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma . \equiv .(\exists y,z).y\in \beta .z\in \gamma .a\downarrow_{,,}y = \alpha \downarrow_{,,}z &\qquad \text{(1)}\\ \vdash . *113·105.*71·57.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \downarrow_{,,}y = \alpha \downarrow_{,,}z. \equiv .y = z:\\ [(1)] \supset :\exists !\alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma . &\equiv .(\exists y,z).y\in \beta .z\in \gamma .y = z.\\ [*13·195] &\equiv .\exists !\alpha \cap \beta \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*113·2. \[\begin{aligned}&\vdash :\xi \in \mu \times _{c} \nu . \equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha \times \beta )\\ &[(*113·03)]\end{aligned}\]

*113·201. \[\begin{aligned}&\vdash \colon\ldotp \xi \in \mu \times _{c} \nu . \equiv :\mu ,\nu \in \text{NC}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\\ &[*113·2.*103·27]\end{aligned}\]

[Pg 116]

*113·202. \(\vdash \colon\ldotp \xi \in \mu \times _{c} \nu . \equiv :\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\xi \text{ sm }(\gamma \times \delta )\)

Dem. \[ \begin{array}{l} \vdash .*113·201.*100·4.\supset \\ \vdash \colon\ldotp \xi \in \mu \times _{c} \nu . &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \in \mu .\beta \in \nu.\xi \text{ sm }(\alpha \times \beta ).\\ [*100·31] &\equiv : (\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\alpha \times \beta ).\\ [*113·13.*73·37] &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\gamma \times \delta ).\\ [*100·31] &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\gamma \times \delta ).\\ [*10·35] &\equiv :\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\xi \text{ sm }(\gamma \times \delta )\colon\ldotp \\ &\supset \vdash .\text{Prop} \end{array} \]

*113·203. \(\vdash :\exists !\mu \times _{c} \nu .\supset .\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C} \quad[*113·201·202·2]\)

*113·204. \(\vdash \colon\ldotp \mu = \Lambda .\lor.\nu = \Lambda .\lor.{\sim}(\mu ,\nu \in \text{NC}):\supset .\mu \times _{c} \nu = \Lambda \quad[*113·203]\)

*113·205. \(\vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu \times _{c} \nu = \Lambda \quad[*113·203]\)

*113·21. \[\begin{aligned}&\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu \times _{c} \nu . \equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\\ &[*113·201]\end{aligned}\]

*113·22. \(\vdash :\xi \in \text{Nc}(\eta )ʻ\gamma \times _{c} \text{Nc}(\zeta )ʻ\delta . \equiv .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\gamma \times \delta )\)

Dem. \[ \begin{array}{l} \vdash .*113·21.*100·41.&\supset \vdash :\xi \in \text{Nc}(\eta )ʻ\gamma \times _{c} \text{Nc}(\zeta )ʻ\delta . \equiv .\\ &(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\alpha \times \beta ).\\ [*102·6] &\equiv .(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\alpha \times \beta ).\\ [*113·13.*73·37] &\equiv .(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\gamma \times \delta ).\\ [*102·6] &\equiv .(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\gamma \times \delta ).\\ [*10·35] &\equiv .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\gamma \times \delta ):\supset \vdash .\text{Prop} \end{array} \]

*113·221. \[\begin{aligned}&\vdash :\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\supset .\text{Nc}(\eta )ʻ\gamma \times _{c} \text{Nc}(\zeta )ʻ\delta = \text{Nc}ʻ(\gamma x \delta )\\ &[*113·22]\end{aligned}\]

*113·222. \(\vdash .\text{N}_{0}\text{c}ʻ\gamma \times _{c} \text{N}_{0}\text{c}ʻ\delta = \text{Nc}ʻ(\gamma \times \delta )\)

Dem. \[ \begin{array}{l} \vdash .*103·1·13.&\supset \vdash .\text{N}_{0}\text{c}ʻ\gamma = \text{Nc}(\gamma )ʻ\gamma .\text{N}_{0}\text{c}ʻ\delta = \text{Nc}(\delta )ʻ\delta .\exists !\text{N}_{0}\text{c}ʻ\gamma .\exists !\text{N}_{0}\text{c}ʻ\delta .\\ [*113·221] &\supset \vdash .\text{N}_{0}\text{c}ʻ\gamma \times _{c} \text{N}_{0}\text{c}ʻ\delta = \text{Nc}ʻ(\gamma x \delta ).\supset \vdash .\text{Prop} \end{array} \]

*113·23. \(\vdash .\mu \times _{c} \nu \in \text{NC}\)

Dem. \[ \begin{array}{l} \vdash .*113·222.*100·41.&\supset \vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\mu \times _{c} \nu \in \text{NC} &\qquad \text{(1)}\\ \vdash .*113·205.*102·74.&\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu \times _{c} \nu \in \text{NC} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*113·24. \(\vdash .\text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta = \text{N}_{0}\text{c}ʻ\gamma \times _{c} \text{N}_{0}\text{c}ʻ\delta \quad[(*113·04·05)]\)

*113·25. \(\vdash .\text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta = \text{Nc}ʻ(\gamma \times \delta )\quad [*113·24·222]\)

[Pg 117]

This proposition constitutes part of the reason for our definitions. It is obvious that such definitions ought, if possible, to be chosen as will yield this proposition.

*113·251. \(\vdash .\gamma \times \delta \in \text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta \quad[*113·25.*100·3]\)

*113·26. \(\vdash :\mu ,\nu \in \text{NC}.\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu \times _{c} \nu = \text{ sm }_{\eta }ʻʻ\mu \times _{c} \text{ sm }_{\zeta }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash .*37·29. \text{Transp}.\supset \vdash :\text{Hp}.&\supset .\exists !\mu .\exists !\nu .\\ [*102·64] &\supset .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta &\qquad \text{(1)}\\ \vdash .*102·88.&\supset \vdash :\mu = \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta .\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\\ &\text{ sm }_{\eta }ʻʻ\mu = \text{Nc}(\eta )ʻ\gamma .\text{ sm }_{\zeta }ʻʻ\nu = \text{Nc}(\zeta )ʻ\delta .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\\ [*113·221]&\supset .\text{ sm }_{\eta }ʻʻ\mu \times _{c} \text{ sm }_{\zeta }ʻʻ\nu = \text{Nc}ʻ(\gamma \times \delta ) &\qquad \text{(2)}\\ \vdash .*37·29.\text{Transp}.*113·221.\supset \\ \vdash :\mu &= \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta .\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu \times _{c} \nu = \text{Nc}ʻ(\gamma \times \delta ) &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash :\mu &= \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta .\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\varsigma }ʻʻ\nu .\supset .\\&\mu \times _{c} \nu = \text{ sm }_{\eta }ʻʻ\mu \times _{c} \text{ sm }_{\varsigma }ʻʻ\nu &\qquad \text{(4)}\\ \vdash .(4).*11·11·35·45.(1).\supset \vdash .\text{Prop} \end{array} \]

*113·261. \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu \times _{c} \nu = \mu ^{(1)} \times _{c} \nu ^{(1)} = \mu _{(00)} \times _{c} \nu _{(00}) = \text{etc}.\)

Here "\(\text{etc}\)." includes all ascending derivatives of \(\mu\). We shall only prove the result for \(\mu ^{(1)}\) and \(\nu ^{(1)}\), since it is proved in just the same way for the other cases. \(\mu^{(1)} \times _{c} \nu ^{(2)}\) or \(\mu ^{(1)} \times _{c} \nu_{(00)}\) or etc. will serve equally well; i.e. it is not necessary to take the same derivative of \(\mu\) as of \(\nu\).

Dem. \[ \begin{array}{l} \vdash .*104·264·265.\supset \\ \vdash :\text{Hp}.\exists !\mu .\exists !\nu .\supset .\mu ^{(1)} = \text{ sm }_{\mu }ʻʻ\mu .\nu ^{(1)} = \text{ sm }_{\nu }ʻʻ\mu .\exists !\mu ^{(1)}.\exists !\nu ^{(1)}.\\ [*113·26] \supset :\mu \times _{c} \nu = \mu ^{(1)} \times _{c} \nu ^{(1)} &\qquad \text{(1)}\\ \vdash .*104·264.*113·204.\supset\\ \vdash :{\sim}(\exists !\mu .\exists !\nu ).\supset .\mu \times _{c} \nu = \Lambda .\mu ^{(1)} \times _{c} \nu ^{(1)} = \Lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

As appears in the above proof, if \(\mu^{i}\) and \(\iota^{j}\) are any derivatives of \(\mu\) and \(\nu\), the above proposition holds provided we have \[ \exists !\mu .\exists !\nu .\supset .\exists !\mu^{i}.\exists !\nu^{j} . \]

Thus it holds for all ascending derivatives, but not always for descending derivatives.

*113·27. \(\vdash .\mu \times _{c} \nu = \nu \times _{c} \mu\)

Dem. \[ \begin{array}{l} \vdash .*113·2·141.\supset \\ \vdash :\xi \in \mu \times _{c} \nu . &\equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\beta \times \alpha ).\\ [*113·2] &\equiv .\xi \in \nu \times _{c} \mu :\supset \vdash .\text{Prop} \end{array} \]

Note that this proposition is not confined to the case in which \(\mu\) and \(\nu\) are cardinals. When either or both are not cardinals, \[ \mu \times _{c} \nu = \Lambda = \nu \times _{c} \mu . \]

[Pg 118]

*113·3. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .\supset .\Sigma ʻ\kappa \in \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*112·24.*113·12.\supset \\ \vdash \colon\ldotp \text{Mult ax}.\exists !\alpha .\supset :\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .&\supset .\Sigma ʻ\kappa \text{ sm }\Sigma ʻ\alpha \downarrow_{,,}ʻʻ\beta .\\ [*113·121]& \supset .\Sigma ʻ\kappa \text{ sm } \beta \times \alpha .\\ [*113·141·25] & \supset .\Sigma ʻ\kappa \in \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta &\qquad \text{(1)}\\ \vdash .*113·114·25.&\supset \vdash :\alpha = \Lambda .\supset .\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta = 0 &\qquad \text{(2)}\\ \vdash .*101·14.\supset \vdash \colon\ldotp \alpha = \Lambda .\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .&\supset :\kappa \in \text{Cl}ʻ\iota ʻ\Lambda :\\ [*60·362] & \supset :\kappa = \iota ʻ\Lambda .\lor.\kappa = \Lambda: \\ [*112·3·301] &\supset :\Sigma ʻ\kappa = \Lambda &\qquad \text{(3)}\\ \vdash .(2).(3).*54·102.&\supset \vdash :\alpha = \Lambda.\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .\supset .\Sigma ʻ\kappa \in \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta &\qquad \text{(4)}\\ \vdash .(1).(4).\supset \vdash .\text{Prop} \end{array} \]

*113·31. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Sigma ʻ\kappa \in \mu \times _{c} \nu \quad[*113·3]\)

*113·32. \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl excl}ʻ\mu .\supset .sʻ\kappa \in \mu \times _{c} \nu \\ &[*112·15.*113·31·23]\end{aligned}\]

*113·33. \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} .\supset :\mu ,\nu \in &\text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .\lambda\in \mu \cap \text{Cl}ʻ\nu .\supset .\\ &\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda = \mu \times _{c} \nu \quad[*113·31·27·23]\end{aligned}\]

*113·34. \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} .\supset :\mu ,\nu \in &\text{NC}.\kappa \in \nu \cap \text{Cl excl}ʻ\mu .\lambda \in \mu \cap \text{Cl excl}ʻ\nu .\supset .\\ &\text{Nc}ʻsʻ\kappa = \text{Nc}ʻsʻ\lambda = \mu \times _{c} \nu \quad[*113·32·27]\end{aligned}\]

The above propositions give the connection of addition and multiplication.

The following propositions are concerned with various forms of the distributive law.

*113·4. \(\vdash .(\beta \cup \gamma ) \times \alpha = (\beta \times \alpha )\cup (\gamma \times \alpha )\)

Dem. \[ \begin{array}{l} \vdash .*113·1.\supset \vdash .(\beta \cup \gamma ) \times \alpha &= sʻ\alpha \downarrow_{,,}ʻʻ(\beta \cup \gamma )\\ [*40·31] & = sʻ\alpha \downarrow_{,,}ʻʻ\beta \cup sʻ\alpha \downarrow_{,,}ʻʻ\gamma \\ [*113·1] &= (\beta \times \alpha )\cup (\gamma \times \alpha ).\supset \vdash .\text{Prop} \end{array} \]

*113·401. \(\vdash :\beta \cap \gamma = \Lambda .\supset .(\beta \times \alpha )\cap (\gamma \times \alpha ) = \Lambda \quad[*113·19.\text{Transp}]\)

*113·41. \[\begin{aligned}\vdash .\text{Nc}ʻ(\beta + \gamma ) \times _{c} \text{Nc}ʻ\alpha = \text{Nc}ʻ\{(\beta + \gamma ) \times \alpha\} &= \text{Nc}ʻ\{(\beta \times \alpha ) + (\gamma \times \alpha )\}\\ &= \text{Nc}ʻ(\beta \times \alpha ) +_{c} \text{Nc}ʻ(\gamma \times \alpha )\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash . *113·25.*110·3. &\supset \vdash .\text{Nc}ʻ(\beta + \gamma ) \times _{c} \text{Nc}ʻ\alpha = \text{Nc}ʻ\{(\beta + \gamma ) \times \alpha\}.\\ &\text{Nc}ʻ\{(\beta \times \alpha ) + (\gamma \times \alpha )\} = \text{Nc}ʻ(\beta \times \alpha ) +_{c} \text{Nc}ʻ(\gamma \times \alpha ) &\qquad \text{(1)}\\ \vdash .*113·4.(*110·01). & \supset \vdash .(\beta + \gamma ) \times \alpha = (\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \times \alpha )\cup (\Lambda _{\beta }\downarrow ʻʻ\iotaʻʻ\gamma \times \alpha ) &\qquad \text{(2)}\\ \vdash .*113·13.*110·12. &\supset \vdash .\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \times \alpha \text{ sm }\beta \times \alpha .\Lambda _{\beta }\downarrow ʻʻ\iotaʻʻ\gamma \times \alpha \text{ sm }\gamma \times \alpha &\qquad \text{(3)}\\ \vdash .*113·401.*110·11. & \supset \vdash .(\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \times \alpha )\cap (\Lambda _{\beta }\downarrow ʻʻ\iotaʻʻ\gamma \times \alpha ) = \Lambda &\qquad \text{(4)}\\ \vdash .*110·152.(2).(3).(4). &\supset \vdash .(\beta + \gamma ) \times \alpha \text{ sm }{(\beta \times \alpha ) + (\gamma \times \alpha )} &\qquad \text{(5)}\\ \vdash .(1).(5).\supset \vdash .\text{Prop} \end{array} \]

[Pg 119]

*113·42. \[\begin{aligned}\vdash .(\text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma ) \times _{c} \text{Nc}ʻ\alpha &= \text{Nc}ʻ(\beta + \gamma ) \times _{c} \text{Nc}ʻ\alpha \\ &= (\text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\alpha ) +_{c} (\text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\alpha )\\ [*110·3.*113·25.*113·41]\end{aligned}\]

*113·421. \[\begin{aligned}\vdash . \text{Nc}ʻ\alpha \times _{c} (\text{Nc}ʻ\beta &+_{c} \text{Nc}ʻ\gamma ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ(\beta + \gamma )\\ &= (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\gamma ) \quad[*113·42·27]\end{aligned}\]

*113·43. \(\vdash .(\nu +_{c} \varpi ) \times _{c} \mu = \mu \times _{c} (\nu +_{c} \varpi ) = (\mu \times _{c} \nu ) +_{c} (\mu \times _{c} \varpi )\)

Dem. \[ \begin{array}{l} \vdash . *113·27·421 .\supset \vdash :\mu ,\nu ,\varpi \in \text{NC}.\supset .(\nu +_{c} \varpi ) \times _{c} \mu &= \mu \times _{c} (\nu +_{c} \varpi )\\ &= (\mu \times _{c} \nu ) +_{c} (\mu \times _{c} \varpi ) &\qquad \text{(1)}\\ \vdash .*113·204.*110·4.\supset\\ \vdash :{\sim}(\mu ,\nu ,\varpi \in \text{NC}).\supset .(\nu +_{c} \varpi ) \times _{c} \mu = \Lambda .\mu \times _{c} &(\nu +_{c} \varpi ) = \Lambda .\\ &(\mu \times _{c} \nu ) +_{c} (\mu \times _{c} \varpi ) = \Lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

The following propositions are concerned with various forms of the distributive law, when the summands are not enumerated, but given as the members of a class.

The first of them (*113·44) gives the distributive law with regard to arithmetical class-multiplication and logical addition of classes.

*113·44. \(\vdash .(sʻ\kappa ) \times \alpha = sʻ(x \alpha )ʻʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*113·1 .\supset \vdash .sʻ(\times \alpha )ʻʻ\kappa &= sʻsʻʻ\alpha \downarrow_{,,}ʻʻʻ\kappa \\ [*42·1] &= sʻsʻ\alpha \downarrow_{,,}ʻʻʻ\kappa \\ [*40·38] &= sʻ\alpha \downarrow_{,,}ʻʻsʻ\kappa \\ [*113·1] &= (sʻ\kappa ) \times \alpha .\supset \vdash .\text{Prop} \end{array} \]

*113·45. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset . \times \alpha ʻʻ\kappa \in \text{Cls}^{2} \text{excl}\)

Dem. \[ \begin{array}{l} \vdash . *113·19. &\supset \vdash :\exists ! \times \alpha ʻ\beta \cap \times \alpha ʻ\gamma .\supset .\exists !\supset \cap \gamma &\qquad \text{(1)}\\ \vdash .(1).*84·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta ,\gamma \in \kappa .\exists ! \times \alpha ʻ\beta \cap \times \alpha ʻ\gamma .&\supset_{\beta,\gamma}.\beta = \gamma .\\ [*30·37] &\supset_{\beta,\gamma}. \times \alpha ʻ\beta = \times \alpha ʻ\gamma :\\ [*37·63] &\supset :\rho ,\sigma \in \times \alpha ʻʻ\kappa .\exists !\rho \cap \sigma .\supset _{\rho ,\sigma}.\rho = \sigma &\qquad \text{(2)}\\ \vdash .(2).*84·11.\supset \vdash .\text{Prop} \end{array} \]

*113·46. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma ʻ \times \alpha ʻʻ\kappa \text{ sm }(\Sigma ʻ\kappa ) \times \alpha\)

Dem. \[ \begin{array}{l} \vdash .*112·15. \supset \vdash :\text{Hp}.&\supset .\Sigma ʻ\kappa \text{ sm }sʻ\kappa .\\ [*113·13] &\supset .(\Sigma ʻ\kappa ) \times \alpha \text{ sm }(sʻ\kappa ) \times \alpha &\qquad \text{(1)}\\ \vdash . *112·15 . *113·45. &\supset \vdash :\text{Hp}.\supset .\Sigma ʻ \times \alpha ʻʻ\kappa \text{ sm }sʻ \times \alpha ʻʻ\kappa &\qquad \text{(2)}\\ \vdash .(1).(2).*113·44.\supset \vdash .\text{Prop} \end{array} \]

[Pg 120]

*113·47. \[\begin{aligned}&\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma \text{Nc}ʻ \times \alpha ʻʻ\kappa = \text{Nc}ʻ{(\Sigma ʻ\kappa ) \times \alpha } = \Sigma \text{Nc}ʻ\kappa \times _{c} \text{Nc}ʻ\alpha \\ &[*113·46]\end{aligned}\]

This is the distributive law for arithmetical multiplication and arithmetical addition of the kind defined in *112.

*113·48. \(\vdash .sʻ\alpha \times ʻʻ\kappa = \alpha \times (sʻ\kappa ) = \text{Cnv}ʻʻ\{(sʻ\kappa ) \times \alpha\}\)

Dem. \[ \begin{array}{l} \vdash .*113·14. \supset \vdash .sʻ\alpha \times ʻʻ\kappa & = sʻ\text{Cnv}ʻʻʻ \times \alpha ʻʻ\kappa \\ [*40·38] &= \text{Cnv}ʻʻsʻ \times \alpha ʻʻ\kappa \\ [*113·44] &= \text{Cnv}ʻʻ{(sʻ\kappa ) \times \alpha } &\qquad \text{(1)}\\ [*113·14] &= \alpha \times sʻ\kappa &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*113·49. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }\alpha \times (\Sigma ʻ\kappa )\)

Dem. \[ \begin{array}{l} \vdash . *113·14.&\supset \vdash .\alpha \times ʻʻ\kappa = \text{Cnv}ʻʻʻ \times \alpha ʻʻ\kappa &\qquad \text{(1)}\\ \vdash .(1).*113·45.*72·11.*84·53.\supset \\ \vdash :\text{Hp}.&\supset .\alpha \times ʻʻ\kappa \in \text{Cls}^{2} \text{excl}.\\ [*112·15] &\supset . \Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }sʻ\alpha \times ʻʻ\kappa .\\ [*113·48] &\supset .\Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }\alpha \times (sʻ\kappa ).\\ [*112·15.*113·13] &\supset .\Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }\alpha \times (\Sigma ʻ\kappa ):\supset \vdash .\text{Prop} \end{array} \]

*113·491. \[\begin{aligned}&\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma \text{Nc}ʻ\alpha \times ʻʻ\kappa = \text{Nc}ʻ{\alpha \times (\Sigma ʻ\kappa )} = \text{Nc}ʻ\alpha \times _{c} \Sigma \text{Nc}ʻ\kappa \\ &[*113·49·25]\end{aligned}\]

The following propositions are concerned with the associative law for arithmetical multiplication.

*113·5. \(\vdash .(\gamma \times \beta ) \times \alpha = \hat{R} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = x\downarrow (y\downarrow z)\}\)

Dem. \[ \begin{array}{l} \vdash .*113·101.\supset \\ \vdash .(\gamma \times \beta ) \times \alpha &= \hat{R} \{(\exists x,P).x\in \alpha .P\in (\gamma \times \beta ).R = x\downarrow P\}\\ [*113·101] &= \hat{R} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = x\downarrow (y\downarrow z)\}.\supset \vdash .\text{Prop} \end{array} \]

*113·51. \(\vdash .(\alpha \times \beta ) \times \gamma \text{ sm }\alpha \times (\beta \times \gamma )\)

Dem. \[ \begin{array}{l} \vdash .*113·141.&\supset \vdash .\alpha \times (\beta \times \gamma )\text{ sm }(\beta \times \gamma ) x \alpha &\qquad \text{(1)}\\ \vdash .*113·5. \supset \vdash .(\alpha \times \beta ) \times \gamma &= \hat{R} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = z\downarrow (y\downarrow x)\} .\\ (\beta \times \gamma ) \times \alpha &= \hat{P} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .P = x\downarrow (z\downarrow y)\} &\qquad \text{(2)}\\ \vdash .(2).\supset \vdash :T &= \hat{R} \hat{P} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = z\downarrow (y\downarrow x).P =x\downarrow (y\downarrow z)\}.\supset .\\ &\text{D}ʻT = (\alpha \times \beta ) \times \gamma .\text{ᗡ}ʻT = (\beta \times \gamma ) \times \alpha &\qquad \text{(3)}\\ \vdash .*21·33.&\supset \vdash :\text{Hp}(3).RTP.RTQ.\supset\\ &(\exists x,x',y,y',z,z').x,x'\in \alpha .y,y'\in \beta .z,z'\in \gamma .R = z\downarrow (y\downarrow x) = z'\downarrow (y'\downarrow x').\\ &P = x\downarrow (z\downarrow y).Q = x'\downarrow (z'\downarrow y').\\ [*55·202]&\supset .P = Q &\qquad \text{(4)}\\ \text{Similarly}\qquad &\vdash :\text{Hp}(3).RTP.QTP.\supset .R = Q &\qquad \text{(5)}\\ \vdash .(3).(4).(5).&\supset \vdash .(\alpha \times \beta ) \times \gamma \text{ sm }(\beta \times \gamma ) \times \alpha &\qquad \text{(6)}\\ \vdash .(1).(6).\supset \vdash .\text{Prop} \end{array} \]

[Pg 121]

*113·511. \(\alpha \times \beta \times \gamma = (\alpha \times \beta ) \times \gamma \quad\text{Df}\)

*113·52. \(\vdash . (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) \times _{c} \text{Nc}ʻ\gamma = \text{Nc}ʻ(\alpha \times \beta \times \gamma ) \quad[*113·25]\)

*113·53. \(\vdash . (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) \times _{c} \text{Nc}ʻ\gamma = \text{Nc}ʻ\alpha \times _{c} (\text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\gamma )\)

Dem. \[ \begin{array}{l} \vdash . *113·52·51 . \supset \\ \vdash . (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) \times _{c} \text{Nc}ʻ\gamma &= \text{Nc}ʻ\{\alpha \times (\beta \times \gamma )\}\\ [*113·25] &= \text{Nc}ʻ\alpha \times _{c} (\text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\gamma ).\supset \vdash .\text{Prop} \end{array} \]

*113·531. \[\begin{aligned}&\vdash . (\text{N}_{0}\text{c}ʻ\alpha \times _{c} \text{N}_{0}\text{c}ʻ\beta ) \times _{c} \text{N}_{0}\text{c}ʻ\gamma = \text{N}_{0}\text{c}ʻ\alpha \times _{c} (\text{N}_{0}\text{c}ʻ\beta \times _{c} \text{N}_{0}\text{c}ʻ\gamma )\\ &[*113·53.(*113·04·05)]\end{aligned}\]

*113·54. \(\vdash . (\mu \times _{c} \nu ) \times _{c} \varpi = \mu \times _{c} (\nu \times _{c} \varpi )\)

Dem. \[ \begin{array}{l} \vdash .*113·531.*103·2.\supset \\ \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .(\mu \times _{c} \nu ) \times _{c} \varpi = \mu \times _{c} (\nu \times _{c} \varpi ) &\qquad \text{(1)}\\ \vdash .*113·204.\supset \\ \vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .(\mu \times _{c} \nu ) \times _{c} \varpi = \Lambda .\mu \times _{c} (\nu \times \varpi ) = \Lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*113·541. \(\mu \times _{c} \nu \times _{c} \varpi = (\mu \times _{c} \nu ) \times _{c} \varpi \quad\text{Df}\)

*113·6. \(\vdash . \text{Nc}ʻ\alpha \times _{c} 0 = 0\)

Dem. \[ \begin{array}{l} \vdash .*113·25.*101·1.\supset \vdash .\text{Nc}ʻ\alpha \times _{c} 0 &= \text{Nc}ʻ(\alpha \times \Lambda )\\ [*113·114.*101·1] &= 0.\supset \vdash .\text{Prop} \end{array} \]

*113·601. \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .\mu \times _{c} 0 = 0\)

Dem. \[ \begin{array}{l} \vdash .*103·26.\supset \vdash :\text{Hp}.\supset .(\exists \alpha ).\mu = \text{N}_{0}\text{c}ʻ\alpha &\qquad \text{(1)}\\ \vdash .*101·11·13.*103·27.\supset \vdash .0 = \text{N}_{0}\text{c}ʻ\Lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash :\text{Hp}.\supset .(\exists \alpha ).\mu \times _{c} 0 = \text{N}_{0}\text{c}ʻ\alpha \times _{c} \text{N}_{0}\text{c}ʻ\Lambda \\ [*113·222] = \text{Nc}ʻ(\alpha x \Lambda )\\ [*113·114.*101·1] = 0:\supset \vdash .\text{Prop} \end{array} \]

[Pg 122]

*113·602. \(\vdash \colon\ldotp \mu \times _{c} \nu = 0. \equiv :\mu ,\nu \in \text{NC} -{℩} ʻ\Lambda :\mu = 0.\lor.\nu = 0\)

Dem. \[ \begin{array}{l} \vdash .*113·203.*101·12.\supset \\ \vdash :\mu \times _{c} \nu = 0.&\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash .(1).*113·201.\supset\\ \vdash \colon\colon \mu \times _{c} \nu = 0.&\supset \colon\ldotp \xi \in 0. \equiv _{\xi } :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\colon\ldotp \\ [*54·102] &\supset \colon\ldotp \xi = \Lambda . \equiv _{\xi } :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\colon\ldotp \\ [*10·1.*13·15] &\supset \colon\ldotp (\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\Lambda \text{ sm }(\alpha \times \beta )\colon\ldotp \\ [*73·47] &\supset \colon\ldotp (\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\alpha \times \beta = \Lambda \colon\ldotp \\ [*113·114] &\supset \colon\ldotp (\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu :\alpha = \Lambda .\lor.\beta = \Lambda \colon\ldotp \\ [*13·195] &\supset \colon\ldotp \Lambda \in \mu .\lor.\Lambda \in \nu \colon\ldotp \\ [(1).*100·45] &\supset \colon\ldotp \mu = \text{Nc}ʻ\Lambda .\lor.\nu = \text{Nc}ʻ\Lambda \colon\ldotp \\ [*101·1] &\supset \colon\ldotp \mu = 0.\lor.\nu = 0 &\qquad \text{(2)}\\ \vdash . *113·601·27. &\supset \vdash \colon\ldotp \mu ,\nu \in \text{NC} - \iota ʻ\Lambda :\mu = 0.\lor.\nu = 0:\supset .\mu \times _{c} \nu = 0 &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

The following propositions are concerned with multiplication by a unit class or by 1 or 2.

*113·61. \(\vdash .\iota ʻz \times \alpha = \downarrow zʻʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*113·1.\supset \vdash .\iota ʻz \times \alpha &= sʻ\alpha \downarrow_{,,}ʻʻ\iota ʻz\\ [*53·31·02] &= \alpha \downarrow_{,,}ʻz\\ [*38·2] &= \downarrow zʻʻ\alpha .\supset \vdash .\text{Prop} \end{array} \]

*113·611. \(\vdash .\iota ʻz \times \alpha \text{ sm }\alpha \quad[*113·61.*73·611]\)

*113·612. \(\vdash .\alpha \times \iota ʻz\text{ sm }\alpha \quad[*113·611·141]\)

*113·62. \(\vdash .\text{Nc}ʻ\alpha \times _{c} 1 = \text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*101·2.\supset \vdash .\text{Nc}ʻ\alpha \times _{c} 1 &= \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\iota ʻz\\ [*113·25] &= \text{Nc}ʻ(\alpha \times \iota ʻz)\\ [*113·612] &= \text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} \end{array} \]

*113·621. \(\vdash :\mu \in \text{NC}.\supset .\mu \times _{c} 1 = \text{ sm }ʻʻ\mu\)

Dem. \[ \begin{array}{l} \vdash .*113·204.\supset \vdash :\mu = \Lambda .\supset .\mu \times _{c} 1 &= \Lambda \\ [*37·29] &= \text{ sm }ʻʻ\mu &\qquad \text{(1)}\\ \vdash .*103·26. \supset \vdash :\text{Hp}.\alpha \in \mu .&\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha . &\qquad \text{(2)}\\ [(*113·04)] \supset .\mu \times _{c} 1 &= \text{Nc}ʻ\alpha \times _{c} 1\\ [*113·62] &= \text{Nc}ʻ\alpha \\ [*103·4.(2)] &= \text{ sm }ʻʻ\mu &\qquad \text{(3)}\\ \vdash .(2).*10·11·23·35.&\supset \vdash :\text{Hp}.\exists !\mu .\supset .\mu \times _{c} 1 = \text{ sm }ʻʻ\mu &\qquad \text{(4)}\\ \vdash .(1).(4).\supset \vdash .\text{Prop} \end{array} \]

[Pg 123]

Observe that if \(\mu\) is a typically definite cardinal, \(\text{ sm }ʻʻ\mu\) is the "same" cardinal rendered typically ambiguous; while if \(\mu\) is typically ambiguous, \(\mu = \text{ sm }ʻʻ\mu\) in every type.

*113·63. \(\vdash :z{\sim}\in \alpha .\supset .\downarrow zʻʻ\alpha \text{ sm } \text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\iota ʻz)\)

Dem. \[ \begin{array}{l} \vdash .*113·152.\supset \vdash :\text{Hp}.\supset .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\iota ʻz)\text{ sm } \alpha \times \iota ʻz &\qquad \text{(1)}\\ \vdash .(1).*113·61·141.\supset \vdash .\text{Prop} \end{array} \]

*113·64. \(\vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm } \alpha \times \beta .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm }\downarrow zʻʻ(\alpha \times \beta )\)

Dem. \[ \begin{array}{l} \vdash .*73·611.*113·13.&\supset \vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm } \alpha \times \beta &\qquad \text{(1)}\\ \vdash .(1).*73·611.&\supset \vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm }\downarrow zʻʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*113·65. \(\vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta = (\downarrow z\parallel \text{Cnv}ʻ\downarrow z)ʻʻ(\alpha \times \beta )\)

Dem. \[ \begin{array}{l} \vdash .*72·184.*55·21.&\supset \vdash .\downarrow z\in 1\rightarrow 1.\alpha \subset \text{ᗡ}ʻ\downarrow z.\beta \subset \text{ᗡ}ʻ\downarrow z.\\ [*113·126] &\supset \vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta = (\downarrow z\parallel \text{Cnv}ʻ\downarrow z)ʻʻ(\alpha \times \beta ).\\ &\supset \vdash .\text{Prop} \end{array} \]

*113·66. \(\vdash .\mu \times _{c} 2 = \mu +_{c} \mu\)

Dem. \[ \begin{array}{l} \vdash .*110·643.\supset \vdash .\mu \times _{c} 2 &= \mu \times _{c} (1 +_{c} 1)\\ [*113·43] &= (\mu \times _{c} 1) +_{c} (\mu \times _{c} 1) &\qquad \text{(1)}\\ \vdash .(1).\supset \vdash \colon\ldotp \mu = \text{N}_{0}\text{c}ʻ\alpha .\supset .\mu \times _{c} 2 &= (\text{N}_{0}\text{c}ʻ\alpha \times _{c} 1) +_{c} (\text{N}_{0}\text{c}ʻ\alpha \times _{c} 1)\\ [*113·62.(*113·04)] &= \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\alpha \\ [*110·3] &= \mu +_{c} \mu &\qquad \text{(2)}\\ \vdash .(2).*103·2. &\supset \vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu \times _{c} 2 = \mu +_{c} \mu &\qquad \text{(3)}\\ \vdash .*113·205.*110·4. &\supset \vdash :\mu {\sim}\in \text{N}_{0}\text{C}.\supset .\mu \times _{c} 2 = \Lambda .\mu +_{c} \mu = \Lambda &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*113·67. \(\vdash .\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ(\beta + \iota ʻy) = (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} \text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*113·421.*101·2.\supset \\ \vdash .\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ(\beta + \iota ʻy) &= (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} (\text{Nc}ʻ\alpha \times _{c} 1)\\ [*113·62] &= (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} \text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} \end{array} \]

*113·671. \(\vdash .\mu \times _{c} (\nu +_{c} 1) = (\mu \times _{c} \nu ) +_{c} \mu \quad[*113·67·205.*110·4]\)

Summary of *114.

The kind of multiplication defined in *113 cannot be extended beyond a finite number of factors. We therefore, as in the case of addition, introduce another definition, defining the product of the numbers of a class of classes, and capable of being applied to an infinite number of factors. We define the product of the numbers of members of \(\kappa\) as \(\text{Nc}ʻ{\in}_{\Delta }ʻ\kappa\); thus we put \[ \Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}. \]

It is to be observed that \(\Pi \text{Nc}ʻ\kappa\) is not a function of \(\text{Nc}ʻʻ\kappa\), because, if two members of \(\kappa\) have the same number, this will count only once in \(\text{Nc}ʻʻ\kappa\), but will count twice in \(\Pi \text{Nc}ʻ\kappa\).

It is very easy to see that, in case \(\kappa\) is finite, \(\text{Nc}ʻ{\in}_{\Delta }ʻ\kappa\) will be what we should ordinarily regard as the product of the numbers of members of \(\kappa\). For suppose (e.g.) \[ \kappa = \iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma , \] where \(\alpha \neq \beta . \alpha \neq \gamma . \beta \neq \gamma\). Then \[ {\in}_{\Delta }ʻ\kappa = \hat{R} \{(\exists x,y,z).R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta \unicode{x228d} z\downarrow \gamma .x\in \alpha .y\in \beta .z\in \gamma\}. \]

Thus if \(R\) is a member of \({\in}_{\Delta }ʻ\kappa\), \(R\) is determinate when \(x\), \(y\), \(z\) are given, \(x\), \(y\), \(z\) being the referents to \(\alpha\), \(\beta\), \(\gamma\). Whether \(\alpha\), \(\beta\), \(\gamma\) overlap or not, the choice of any one of \(x\), \(y\), \(z\) is entirely independent of the choice of the other two, and therefore the total number of choices possible is obviously the product of the numbers of \(\alpha\), \(\beta\), \(\gamma\). Thus our definition will not conflict with what is commonly understood by a product.

The propositions of this number are less numerous and less important than those of *113. We shall deal first with products of a single factor, and products in which one factor is null (*114·2—·27). We shall then deal (*114·3—·36) with the relations between the sort of multiplication here defined and the sort defined in *113. Then we have a few propositions (*114·4—*114·43) showing that unit factors make no difference to the value of a product. Then we prove (*114·5—·52) that the value of the product is the same for two classes having double similarity, and then (*114·53—·571) we give extensions of this result which depend upon the multiplicative axiom. Finally, we give some new forms of the associative law of multiplication.

[Pg 125]

Among the more important propositions in this number are the following:

*114·21. \(\vdash . \Pi \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha\)

I.e. a product of one factor is equal to that factor.

*114·23. \(\vdash :\Lambda \in \kappa .\supset .\Pi \text{Nc}ʻ\kappa = 0\)

I.e. a product vanishes if one of its factors is zero. The converse requires the multiplicative axiom, as appears from the proposition

*114·26. \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0. \equiv _{\kappa } .\Lambda \in \kappa\)

I.e. the multiplicative axiom is equivalent to the assumption that a product vanishes when, and only when, one of its factors is zero.

*114·301. \(\vdash :\kappa \cap \lambda = \lambda .\supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm } {\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta} ʻ\lambda \)

whence

*114·31. \(\vdash :\kappa \cap \lambda = \Lambda .\supset .\Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ(\kappa \cup \lambda )\)

which is a form of the associative law, and

*114·35. \(\vdash :\alpha \neq \beta .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta\)

which connects the two sorts of multiplication.

*114·41. \(\vdash :\lambda \subset 1.\supset .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) = \Pi \text{Nc}ʻ\kappa\)

I.e. unit factors make no difference to the value of a product.

*114·51. \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\)

This proposition gives a correlator of \({\in}_{\Delta }ʻ\kappa\) and \({\in}_{\Delta }ʻ\lambda\) as a function of a double correlator of \(\kappa\) and \(\lambda\), and thus leads to

*114·52. \(\vdash :\kappa \text{ sm }\text{ sm }\lambda .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda .{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\lambda\)

Hence, by the propositions of *111, we infer

*114·571. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda\)

I.e. assuming the multiplicative axiom, if \(\kappa\) and \(\lambda\) each consist of \(\mu\) classes of \(\nu\) terms each, their products are equal.

We have next various forms of the associative law, beginning with

*114·6. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \Pi \text{Nc}ʻ{\in}_{\Delta }ʻʻ\kappa = \Pi \text{Nc}ʻsʻ\kappa \)

which is an immediate consequence of *85·44. The other form is

*114·632. \[\begin{aligned}\vdash :S\upharpoonright &\gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\gamma \cap Sʻʻ\gamma = \Lambda .\supset .\\ &{\in}_{\Delta }ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma )\end{aligned}\]

As to the sense in which this is a form of the associative law, see the observations following *114·6.

*114·01. \(\Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ{\in}_{\Delta }ʻ\kappa \quad\text{Df}\)

*114·1. \(\vdash . \Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad[(*114·01)]\)

*114·11. \(\vdash :\beta \in \Pi \text{Nc}ʻ\kappa . \equiv .\beta \text{ sm }{\in}_{\Delta}ʻ\kappa . \equiv .\beta \in \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad[*114·1.*100·31]\)

*114·12. \(\vdash .{\in}_{\Delta}ʻ\kappa \in \Pi \text{Nc}ʻ\kappa \quad[*100·3.*114·1]\)

*114·2. \(\vdash .\Pi \text{Nc}ʻ\Lambda = 1 \quad[*83·15 .*101·2]\)

[Pg 126]

Thus a product of no factors is 1. This is the source of \(\mu^{0} =1\), as we shall see later.

*114·21. \(\vdash .\Pi \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha \quad[*83·41]\)

*114·22. \(\vdash . \Pi \text{Nc}ʻ\iota ʻ\Lambda = 0 \quad[*114·21.*101·1]\)

*114·23. \(\vdash :\Lambda \in \kappa .\supset .\Pi \text{Nc}ʻ\kappa = 0 \quad[*83·11.*101·1]\)

Thus an arithmetical product is zero if any of its factors is zero. To prove the converse, we have to assume the multiplicative axiom, which, in fact, is equivalent to the proposition that an arithmetical product is only zero when at least one of its factors is zero.

*114·24. \(\vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\kappa \subset \lambda .\supset .\Pi \text{Nc}ʻ\kappa \neq 0\)

Dem. \[ \begin{array}{l} \vdash .*114·1.*101·1. &\supset \vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\supset .\exists !{\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ \vdash . (1) . *80·6. \supset \vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\kappa \subset \lambda .&\supset .\exists !{\in}_{\Delta}ʻ\kappa .\\ [*114·1.*101·1] &\supset .\Pi \text{Nc}ʻ\kappa \neq 0:\supset \vdash .\text{Prop} \end{array} \]

*114·25. \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0.\supset _{\kappa }.\Lambda \in \kappa\)

Dem. \[ \begin{array}{l} \vdash .*88·37.\text{Transp}.\supset \\ \vdash \colon\ldotp \text{Mult ax}. &\equiv :{\in}_{\Delta}ʻ\kappa = \Lambda .\supset _{\kappa }.\Lambda \in \kappa :\\ [*114·1.*101·1] &\equiv :\Pi \text{Nc}ʻ\kappa = 0.\supset _{\kappa }.\Lambda \in \kappa \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

Note that \(\Lambda \in \kappa . \equiv .0\in \text{Nc}ʻʻ\kappa\).

*114·26. \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0. \equiv _{\kappa } .\Lambda \in \kappa \quad[*88·372.*101·1]\)

*114·261. \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0. \equiv _{\kappa } .0\in \text{Nc}ʻʻ\kappa \quad[*114·26.*101·1]\)

*114·27. \[\begin{aligned}&\vdash \colon\colon \text{Mult ax}. \equiv \colon\ldotp \alpha \in \kappa .\supset _{\alpha }.\exists !\alpha : \equiv _{\kappa } .\Pi \text{Nc}ʻ\kappa \neq 0\\ &[*114·26. \text{Transp}.*24·63]\end{aligned}\]

*114·3. \(\vdash :\kappa \neq \lambda .\supset .{\in}_{\Delta}ʻ(\iota ʻ{\in}_{\Delta}ʻ\kappa \cup \iota ʻ{\in}_{\Delta}ʻ\lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*113·146. \supset \vdash :{\in}_{\Delta}ʻ\kappa \neq {\in}_{\Delta}ʻ\lambda .\supset .{\in}_{\Delta}ʻ(\iota ʻ{\in}_{\Delta}ʻ\kappa \cup \iota ʻ{\in}_{\Delta}ʻ\lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ \vdash .*80·81. \supset \vdash \colon\ldotp \exists !{\in}_{\Delta}ʻ\kappa .\lor.\exists !{\in}_{\Delta}ʻ\lambda :\kappa \neq \lambda :\supset .{\in}_{\Delta}ʻ\kappa \neq {\in}_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ \vdash .*83·903.*113·114.\supset \\ \vdash :{\in}_{\Delta}ʻ\kappa = \Lambda .{\in}_{\Delta}ʻ\lambda = \Lambda .\supset .{\in}_{\Delta}ʻ(\iota ʻ{\in}_{\Delta}ʻ\kappa \cup \iota ʻ{\in}_{\Delta}ʻ\lambda ) = \Lambda .{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda = \Lambda &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*114·301. \(\vdash :\kappa \cap \lambda = \Lambda .\supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*85·45.*114·3.\supset\\ \vdash :\kappa \cap \lambda = \Lambda .\kappa \neq \lambda .\supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ \vdash .*22·5. \supset \vdash :\kappa \cap \lambda = \Lambda .\kappa = \lambda .\supset .\kappa = \Lambda .\lambda = \Lambda .\\ [*83·15] \supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda ) = \iota ʻ\dot{\Lambda} .{\in}_{\Delta}ʻ\kappa = \iota ʻ\dot{\Lambda} .{\in}_{\Delta}ʻ\lambda = \iota ʻ\dot{\Lambda} .\\ [*113·611] \supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 127]

*114·31. \[\begin{aligned}&\vdash :\kappa \cap \lambda = \Lambda .\supset .\Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ(\kappa \cup \lambda )\\ &[*114·301·1.*113·25]\end{aligned}\]

The above is one form of the associative law of multiplication.

*114·311. \(\vdash .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) = \Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ(\lambda - \kappa ) \quad[*114·31.*22·91]\)

*114·32. \(\vdash :\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0. \equiv .\Pi \text{Nc}ʻ\kappa \neq 0.\Pi \text{Nc}ʻ\lambda \neq 0\)

Dem. \[ \begin{array}{l} \vdash .*114·311.*113·602.\supset \\ \vdash :\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0.\supset .\Pi \text{Nc}ʻ\kappa \neq 0 &\qquad \text{(1)}\\ \vdash .(1) \frac{\lambda ,\kappa}{\kappa ,\lambda}. \supset \vdash :\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0.\supset .\Pi \text{Nc}ʻ\lambda \neq 0 &\qquad \text{(2)}\\ \vdash .*114·24.\supset \vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\supset .\Pi \text{Nc}ʻ(\lambda - \kappa ) \neq 0:\\ [\text{Fact}] \supset \vdash :\Pi \text{Nc}ʻ\kappa \neq 0.\Pi \text{Nc}ʻ\lambda \neq 0.\supset .\Pi \text{Nc}ʻ\kappa \neq 0.\Pi \text{Nc}ʻ(\lambda -\kappa ) \neq 0.\\ [*113·602.*114·311] \supset .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0 &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*114·33. \(\vdash :\alpha {\sim}\in \kappa .\supset .\Pi \text{Nc}ʻ(\kappa \cup \iota ʻ\alpha ) = \Pi \text{Nc}ʻ\kappa \times _{c} \text{Nc}ʻ\alpha \quad[*114·31·21]\)

*114·34. \[\begin{aligned}&\vdash :\Pi \text{Nc}ʻ\kappa \neq 0.\exists !\alpha . \equiv .\Pi \text{Nc}ʻ(\kappa \cup \iota ʻ\alpha ) \neq 0\\ &[*114·32·21.*101·14]\end{aligned}\]

*114·35. \(\vdash :\alpha \neq \beta .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta \quad[*114·33·21]\)

*114·36. \[\begin{aligned}&\vdash :\alpha \neq \beta .\alpha \neq \gamma .\beta \neq \gamma .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\gamma \\ &[*114·33·35]\end{aligned}\]

*114·4. \(\vdash :\lambda \subset 1.\supset .\Pi \text{Nc}ʻ\lambda = 1 \quad[*83·44]\)

*114·41. \(\vdash :\lambda \subset 1.\supset .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) = \Pi \text{Nc}ʻ\kappa \quad[*83·57]\)

*114·42. \(\vdash .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ(\kappa -1)\)

Dem. \[ \begin{array}{l} \vdash .*24·41.\supset \vdash .\kappa = (\kappa - 1)\cup (\kappa \cap 1) &\qquad \text{(1)}\\ \vdash .(1).*114·41.\supset \vdash .\text{Prop} \end{array} \]

*114·43. \(\vdash .\Pi \text{Nc}ʻ(\kappa \cup \iotaʻʻ\alpha ) = \Pi \text{Nc}ʻ\kappa \quad[*114·41.*52·3]\)

*114·5. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\)

Dem. \[ \begin{array}{l} \vdash .*111·1·11. &\supset \vdash :\text{Hp}.\supset .T,T_{\in }\upharpoonright \lambda \in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash .*80·14.*83·21.&\supset \vdash sʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\lambda \subset sʻ\lambda .sʻ\text{ᗡ}ʻʻ{\in}_{\Delta}ʻ\lambda \subset \lambda &\qquad \text{(2)}\\ \vdash .(1).(2).*74·773.\supset \\ \vdash :\text{Hp}.&\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda \in \{(T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda \},\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda ) &\qquad \text{(3)}\\ \vdash .*82.43 \frac{\in,T_{\in}}{P,\,Q}.*62·3.\supset \\ \vdash :T,T_{\in }\upharpoonright \lambda \in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\lambda \subset \text{ᗡ}ʻT_{\in }.\kappa &= T_{\in }ʻʻ\lambda .\supset .\\ &(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in })_{\Delta }ʻ\kappa = (T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(4)}\\ \vdash.(4).(1).*111·1.*37·111.\supset \vdash :\text{Hp}.\supset .&(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in })_{\Delta }ʻ\kappa = (T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(5)}\\ \vdash .*34·1.*37·101.\supset \\ \vdash :x(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in }\alpha . &\equiv .(\exists y,\beta ).xTy.y\in \beta .\beta \in \lambda .\alpha = Tʻʻ\beta &\qquad \text{(6)}\\ \vdash .(6).*72·52.*111·1.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :x(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in })\alpha . &\equiv .(\exists y,\beta ).xTy.y\in \beta .\beta \in \lambda .\beta = \breve{T} ʻʻ\alpha .\alpha \subset \text{D}ʻT.\\ [*111·1·131.*13·195] &\equiv .(\exists y).xTy.y\in \breve{T} ʻʻ\alpha .\alpha \in \kappa .\\ [*37·1] &\equiv .x\in Tʻʻ\breve{T} ʻʻ\alpha .a\in \kappa .\\ [*72·502.*111·1] &\equiv .x(\in \upharpoonright \kappa )\alpha &\qquad \text{(7)}\\ \vdash .(5).(7).\supset \vdash :\text{Hp}.&\supset .(\in \upharpoonright \kappa )_{\Delta }ʻ\kappa = (T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda .\\ [*83·12] &\supset .{\in}_{\Delta}ʻ\kappa = (T \parallel ]\breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(8)}\\ \vdash .(3).(8).\supset \vdash .\text{Prop} \end{array} \]

[Pg 128]

*114·501. \(\vdash :S = T\upharpoonright sʻ\lambda .\supset .(S\parallel \breve{S} _\in )\upharpoonright {\in}_{\Delta}ʻ\lambda = (T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*80·14.*83·21.\supset \\ \vdash \colon\ldotp R\in {\in}_{\Delta}ʻ\lambda .\supset :yR\beta .&\supset .y\in sʻ\lambda .\beta \in \lambda .\\ [*40·13] &\supset .y\in sʻ\lambda .\beta \subset sʻ\lambda : &\qquad \text{(1)}\\ [*4·71 . \text{Fact}] &\supset :xTy.yR\beta .\beta \breve{T} _{\in }\alpha . \equiv .xTy.y\in sʻ\lambda .yR\beta .\beta \breve{T} _{\in }\alpha .\beta \subset sʻ\lambda .\\ [*37·101.*22·621] &\equiv .x(T\upharpoonright sʻ\lambda )y.yR\beta .\alpha = Tʻʻ\beta .\beta = \beta \cap sʻ\lambda .\\ [(1).*37·412] &\equiv .x(T\upharpoonright sʻ\lambda )y.yR\beta .\alpha = (T\upharpoonright sʻ\lambda )ʻʻ\beta &\qquad \text{(2)}\\ \vdash .(2).\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\in {\in}_{\Delta}ʻ\kappa .\supset .T\mid R\mid \breve{T} _{\in } = S\mid R\mid \breve{S} _{\in }:\\ [*35·71] &\supset :(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\kappa = (S\parallel \breve{S} _{\in })\upharpoonright {\in}_{\Delta}ʻ\kappa \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*114·51. \[\begin{aligned}&\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\\ &[*114·5·501]\end{aligned}\]

*114·52. \(\vdash :\kappa \text{ sm }\text{ sm } \lambda .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda .{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\lambda \quad[*114·51 .*111·4]\)

*114·53. \[\begin{aligned}\vdash \colon\colon &\text{Mult ax}.\supset \colon\ldotp \kappa ,\lambda \in \text{Cls}^{2} \text{excl}:\\ &(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda :\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda\\ &[*114·52.*111·5]\end{aligned}\]

*114·54. \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl excl}ʻ\nu .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \\ &[*114·52.*111·53]\end{aligned}\]

The condition \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\), which is involved in the hypothesis of *114·54 (through \(\kappa\), \(\lambda \in \text{Cl excl}ʻ\nu\)), is not necessary. The following propositions enable us to remove it. We first prove \[ {\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\in \unicode{x21A7}ʻʻ\kappa \] and then we use *114·54 to take us from \({\in}_{\Delta}ʻ\in\unicode{x21A7}ʻʻ\kappa\) to \({\in}_{\Delta}ʻ\in\unicode{x21A7}ʻʻ\lambda \). Thence we arrive at \({\in}_{\Delta}ʻ\kappa \text{ sm } {\in}_{\Delta}ʻ\lambda\).

*114·56. \(\vdash .{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\in \unicode{x21A7}ʻʻ\kappa .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*85·54]\)

*114·561. \[\begin{aligned}&\vdash :S\in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.\supset .\in \unicode{x21A7}\mid S\mid \text{Cnv}ʻ(\in \unicode{x21A7})\in (\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, (e\unicode{x21A7}ʻʻ\lambda )\cap \text{ᗡ}ʻʻʻ\text{ sm }\\ &[*73·63.*85·601.*38·12.*33·432]\end{aligned}\]

[Pg 129]

*114·562. \(\vdash \colon\ldotp \text{Mult ax}.\supset : (\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset .\in \unicode{x21A7}ʻʻ\kappa \text{ sm }\text{ sm }\in \unicode{x21A7}ʻʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash .*114·561.*85·61.\supset \\ \vdash \colon\ldotp (\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset :\\ \in \unicode{x21A7}ʻʻ\kappa ,\in \unicode{x21A7}ʻʻ\lambda \in \text{Cls}^{2} \text{excl}:(\exists T).T\in 1\rightarrow 1.T\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻT = \in \unicode{x21A7}ʻʻ\kappa .\text{ᗡ}ʻT = \in \unicode{x21A7}ʻʻ\lambda :\\ [*111·5] \supset :\text{Mult ax}.\supset .\in \unicode{x21A7}ʻʻ\kappa \text{ sm } \text{ sm }\in \unicode{x21A7}ʻʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*114·57. \[\begin{aligned}\vdash \colon\ldotp &\text{Mult ax}.\supset :\\ &(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*114·562·52.\supset \\ \vdash \colon\ldotp \text{Mult ax}.\supset :(\exists S).&S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset .\\ &\Pi \text{Nc}ʻ\in \downarrow ʻʻ\kappa = \Pi \text{Nc}ʻ\in \downarrow ʻʻ\lambda .\\ [*114·56] &\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*114·571. \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \\ &[*111·52 . *114·57]\end{aligned}\]

*114·6. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Pi \text{Nc}ʻ{\in}_{\Delta}ʻʻ\kappa = \Pi \text{Nc}ʻsʻ\kappa \quad[*85·44]\)

This is the most general form of the associative law for arithmetical multiplication.

Owing to the fact that we have two kinds of multiplication, namely \(\alpha \times \beta\) and \({\in}_{\Delta}ʻ\kappa\), we have four forms of the associative law of multiplication, namely:

(1) *114·6, above,

(2) *113·54, i.e. \(\vdash .(\mu \times _{c} \nu ) \times _{c}\varpi = \mu \times _{c} (\nu \times _{c} \varpi )\),

(3) *114·31, i.e. \(\vdash :\kappa \cap \lambda = \Lambda.\supset .\Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ(\kappa \cup \lambda )\),

(4) a form of the associative law which has not yet been proved, which may be explained as follows.

Suppose we have a number of pairs of classes, e.g. \((\alpha_{1}, \beta _{1})\), \((\alpha _{2}, \beta _{2})\), \((\alpha _{3},\beta _{3})\),.... Suppose we form the products \(\alpha _{1} \times\beta _{1}\), \(\alpha _{2} \times \beta _{2}\), \(\alpha _{3} \times\beta _{3}\), ... and multiply all these products together. We wish to prove that (with a suitable hypothesis) the result is similar to the product of all the \(\alpha\)'s and all the \(\beta\)'s taken together as one class; i.e. if we call \(\lambda\) the class of products \(\alpha _{1} \times \beta _{1}\), \(\alpha _{2} \times \beta _{2}\), \(\alpha _{3} \times \beta _{3}\), ..., and \(\mu\) the class whose members are \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), ..., \(\beta _{1}\), \(\beta _{2}\), \(\beta _{3}\), ..., we wish to prove \[ \Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ\mu . \] In order to express this proposition in symbols, let \(S\) be the correlator of the \(\alpha\)'s and \(\beta\)'s, so that \(\beta_{\nu} =Sʻ\alpha_{\nu}\). (The suffix \(\nu\) will not be used further, since it implies that the number of \(\alpha\)'s and of \(\beta\)'s is finite or denumerable.) Then our class of products of the form \(\alpha \times \beta\) is \[ \hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}, \][Pg 130] where \(\gamma\) is the class of all the \(\alpha\)'s ; and the product of this class of products is \[ {\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}. \] On the other hand, the class of all the \(\alpha\)'s and \(\beta\)'s is \(\gamma \cup Sʻʻ\gamma\), and the product of this class is \[ {\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma ). \] Thus what we have to prove (with a suitable hypothesis) is \[ {\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\} \text{ sm } {\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma ). \] The hypothesis required is \[ S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\gamma \cap Sʻʻ\gamma = \Lambda . \]

A smaller hypothesis suffices, however, for a proposition which, in virtue of *114·301, is closely allied to the above, namely \[ {\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}. \] For this, a sufficient hypothesis is \[ S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS. \] Thus e.g. we may write \(I\) for \(S\), and we find \[ \vdash .{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻ\gamma \text{ sm } {\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times \alpha\}. \]

We shall now prove the above propositions. What follows, down to *114·621, consists of lemmas.

For convenience, we write \(S_{\times }ʻ\alpha\) for \(\alpha \times Sʻ\alpha\) in the course of these lemmas; this notation is introduced in the hypotheses of the lemmas.

*114·601. \[\begin{aligned}\vdash \colon\ldotp S\upharpoonright \gamma &\in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\Lambda {\sim}\in \gamma .S_{\times } = \hat{\mu} \hat{\alpha} (\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha ).\supset :\\ &S_{\times }\in 1\rightarrow 1.\text{ᗡ}ʻS_{\times } = \gamma .\text{D}ʻS_{\times } = \hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}:\\ &\alpha \in \gamma .\supset _{\alpha }.S_{\times }ʻ\alpha = \alpha \times Sʻ\alpha\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*33·11. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻS_{\times } = \hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\} &\qquad \text{(1)}\\ \vdash .*21·33 \supset \vdash \colon\ldotp \text{Hp}.\alpha \in \gamma .&\supset :\mu (S_{\times })\alpha . \equiv _{\mu } .\mu = \alpha \times Sʻ\alpha :\\ [*30·3] &\supset :S_{\times }ʻ\alpha = \alpha \times Sʻ\alpha &\qquad \text{(2)}\\ \vdash .(2).*14·204. \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \gamma .&\supset _{\alpha }.\text{E}!S_{\times }ʻ\alpha . &\qquad \text{(3)}\\ [*33·43] &\supset _{\alpha }.\alpha \in \text{ᗡ}ʻS_{\times } &\qquad \text{(4)}\\ \vdash .*21·33.*33·131.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{ᗡ}ʻS_{\times }.\supset _{\alpha }.\alpha \in \gamma &\qquad \text{(5)}\\ \vdash .(4).(5). \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻS_{\times } = \gamma . &\qquad \text{(6)}\\ [(3).*71·16] &\supset .S_{\times }\in 1\rightarrow \text{Cls} &\qquad \text{(7)}\\ \vdash .*113·181. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\alpha'\in \gamma .\alpha \times Sʻ\alpha = \alpha'\times Sʻ\alpha'.\supset .Sʻ\alpha = Sʻ\alpha'.\\ [*71·59] &\supset .\alpha = \alpha' &\qquad \text{(8)}\\ \vdash .(8).*71·55.(2).(6).(7).&\supset \vdash :\text{Hp}.\supset .S_{\times }\in 1\rightarrow 1 &\qquad \text{(9)}\\ \vdash .(1).(2).(6).(9).\supset \vdash .\text{Prop} \end{array} \]

[Pg 131]

*114·602. \(\vdash :\text{Hp}*114·601.A = \hat{R} \hat{\alpha} \{\alpha \in \gamma .R = (Sʻ\alpha )\downarrow \alpha\}.\supset .A\in 1\rightarrow 1.\text{ᗡ}ʻA = \gamma\)

Dem.

As in *114·601, we prove

\[ \begin{array}{l} \vdash :\text{Hp}.&\supset .A\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻA = \gamma &\qquad \text{(1)}\\ \vdash .*21·33.*13·171.\supset \vdash \colon\ldotp \text{Hp}.\supset :RA\alpha .RA\beta .&\supset .(Sʻ\alpha )\downarrow \alpha = (Sʻ\beta )\downarrow \beta .\\ [*55·202] &\supset .\alpha = \beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*114·603. \(\vdash :\text{Hp}*114·602.X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma .P = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }.\supset .P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times }\)

Dem. \[ \begin{array}{l} \vdash .*43·122.*71·166.*144·601·602.&\supset \vdash :\text{Hp}.\supset .P\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash .*43·122.*37·32·322.*33·431.\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻP &= S_{\times }ʻʻ\text{ᗡ}ʻA\\ [*114·601·602] &= \text{D}ʻS_{\times } &\qquad \text{(2)}\\ \vdash .*34·1.\supset \colon\ldotp \text{Hp}.\supset :\\ MP\mu . &\equiv .(\exists R,\alpha ).M = Y\mid R\mid \breve{X} .R = (Sʻ\alpha )\downarrow \alpha .\alpha \in \gamma .\mu = S_{\times }ʻ\alpha .\\ [*113·123.*80·14] &\equiv .(\exists \alpha ).M = (YʻSʻ\alpha )\downarrow (Xʻ\alpha ).\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ [*13·195.*114·601] &\equiv .(\exists \alpha ,\beta ).\beta = Sʻ\alpha .\alpha \in \gamma .M = (Yʻ\beta )\downarrow (Xʻ\alpha ).\mu = \alpha \times \beta .\\ [*83·2] &\supset .(\exists \alpha ,\beta ,u,v).\beta = Sʻ\alpha .\alpha \in \gamma .u\in \alpha .v\in \beta .\\ &M = (v\downarrow u).\mu = \alpha \times \beta .\\ [*113·101] &\supset .M\in \mu &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*80·14.\supset \vdash .\text{Prop} \end{array} \]

*114·604. \[\begin{aligned}\vdash :\text{Hp}*114·602.T = &\hat{P} \hat{Q}\{(\exists X,Y).X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma\}.\\ &\{Q = Y\downarrow X.P = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }\}.\\ &\supset .T\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻT = {\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma .\text{D}ʻT\subset {\in}_{\Delta}ʻ\text{D}ʻS_{\times }\end{aligned}\]

The relation \(T\) here defined is the correlator required for proving \[ {\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma . \]

Besides what is proved in the present proposition, we shall have to prove \[ T\in \text{Cls}\rightarrow 1.{\in}_{\Delta}ʻ\text{D}ʻS_{\times }\subset \text{D}ʻT. \]

The proof of the present proposition is as follows.

Dem. \[ \begin{array}{l} \vdash .*21·33.*13·171.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ PTQ.P'TQ.&\supset .(\exists X,Y,X',Y').Y\downarrow X = Y'\downarrow X'.P = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }.\\ &P' = (Y'\parallel \breve{X}')\mid A\mid \breve{S} _{\times }.\\ [*55·202] &\supset .P = P' &\qquad \text{(1)}\\ \vdash .*21·33.*114·603.&\supset \vdash \colon\ldotp \text{Hp}.\supset :PTQ.\supset .P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times } &\qquad \text{(2)}\\ \vdash .(1).(2).*113·101.\supset \vdash .\text{Prop} \end{array} \]

[Pg 132]

*114·605. \(\vdash :\text{Hp}*114·604.\supset .T\in \text{Cls}\rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash .*114·601.&\supset \vdash :\text{Hp}.\supset .S_{\times }\in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash .(1).*74·71.*114·601·602.\supset \\ \vdash \colon\ldotp \text{Hp}.X,X'\in {\in}_{\Delta}ʻ\gamma .&Y,Y'\in {\in}_{\Delta}ʻSʻʻ\gamma .(Y\parallel \breve{X})\mid A\mid \breve{S} _{\times } = (Y'\parallel \breve{X}')\mid A\mid \breve{S} _{\times }.\supset :\\ &(Y\parallel \breve{X} )\mid A = (Y'\parallel \breve{X}')\mid A:\\ [*74·7] &\supset :(Y\parallel \breve{X} )\upharpoonright \text{D}ʻA = (Y'\parallel \breve{X}')\upharpoonright \text{D}ʻA:\\ [*114·602] \supset :\alpha \in \gamma .&\supset _{\alpha }.(Y\parallel \breve{X})ʻ(Sʻ\alpha )\downarrow \alpha = (Y'\parallel \breve{X}')ʻ(Sʻ\alpha )\downarrow \alpha .\\ [*113·123] &\supset _{\alpha }.(YʻSʻ\alpha )\downarrow (Xʻ\alpha ) = (Y'ʻSʻ\alpha )\downarrow (X'ʻ\alpha ).\\ [*55·202] &\supset _{\alpha }.Xʻ\alpha = X'ʻ\alpha .YʻSʻ\alpha = Y'ʻSʻ\alpha :\\ [*80·14.*33·45] &\supset :X = X'.Y = Y':\\ [*55·202] &\supset :Y\downarrow X = Y'\downarrow X' &\qquad \text{(2)}\\ \vdash . (2) . *13·22. *21·33. \supset \vdash \colon\ldotp \text{Hp}.&\supset :PTQ.PTQ'.\supset .Q = Q'\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

The following propositions are required for proving that, with the same hypothesis, \({\in}_{\Delta}ʻ\text{D}ʻS_{\times }\subset \text{D}ʻT\).

*114·61. \[\begin{aligned}\vdash :\text{Hp}*114·602.P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times }.&X = \breve{\iota} \mid \text{ᗡ}\mid P\mid S_{\times }.Y = \breve{\iota} \mid \text{D}\mid P\mid S_{\times }\mid \breve{S} .\supset .\\ &X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*72·181·13·131.*80·14.*114·601.\supset \vdash :\text{Hp}.\supset .X,Y\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash .*72·2·181·13·131.*80·14.*114·601.\supset\\ \vdash \colon\ldotp \text{Hp}.\supset :xX\alpha . \equiv .x = \breve{\iota} ʻ\text{ᗡ}ʻPʻS_{\times }ʻ\alpha . &\qquad \text{(2)}\\ [*51·53] \supset .x\in \text{ᗡ}ʻPʻS_{\times }ʻ\alpha .\\ [*83·2. *114·601] \supset .(\exists R).R\in \alpha \times Sʻ\alpha .x\in \text{ᗡ}ʻR.\\ [*113·142] \supset .x\in \alpha &\qquad \text{(3)}\\ \vdash . *114·601. \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \gamma . \equiv .S_{\times }ʻ\alpha \in \text{D}ʻS_{\times }.\\ [*83·2] \equiv .\text{E}!PʻS_{\times }ʻ\alpha &\qquad \text{(4)}\\ \vdash . *83·2. \supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!PʻS_{\times }ʻ\alpha . \equiv .PʻS_{\times }ʻ\alpha \in S_{\times }ʻ\alpha .\\ [*113·142] \supset .\text{ᗡ}ʻPʻS_{\times }ʻ\alpha \in 1.\\ [*52·15] \supset .\text{E}!\breve{\iota} ʻ\text{ᗡ}ʻPʻS_{\times }ʻ\alpha &\qquad \text{(5)}\\ \vdash .(2).(4).(5). \supset \vdash :\text{Hp}.\supset .\gamma \subset \text{ᗡ}ʻX &\qquad \text{(6)}\\ \vdash . *34·36. *114·601. \supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻX\subset \gamma &\qquad \text{(7)}\\ \vdash .(1).(3).(6).(7). \supset \vdash :\text{Hp}.\supset .X\in {\in}_{\Delta}ʻ\gamma &\qquad \text{(8)}\\ \text{Similarly}\qquad \vdash :\text{Hp}.\supset .Y\in {\in}_{\Delta}ʻSʻʻ\gamma &\qquad \text{(9)}\\ \vdash .(8).(9).\supset \vdash .\text{Prop} \end{array} \]

*114·611. \(\vdash \colon\ldotp \text{Hp}*114·61.\supset :\alpha \in \gamma .\supset .(YʻSʻ\alpha )\downarrow (Xʻ\alpha ) = PʻS_{\times }ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*72·2. \supset \vdash :\text{Hp}.\alpha \in \gamma .&\supset .Xʻ\alpha = \breve{\iota} ʻ\text{ᗡ}ʻPʻS_{\times }ʻ\alpha .YʻSʻ\alpha = \breve{\iota} ʻ\text{D}ʻPʻS_{\times }ʻ\alpha .\\ [*55·16. *51·51] &\supset .(YʻSʻ\alpha )\downarrow (Xʻ\alpha ) = PʻS_{\times }ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

[Pg 133]

*114·612. \(\vdash :\text{Hp} *114·61.\supset .(Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times } = P\)

Dem. \[ \begin{array}{l} \vdash .*83·15.\supset \vdash :\text{Hp}.\dot{\exists} !P.&\supset .\exists !\text{D}ʻS_{\times } .\\ [*114·601] &\supset .\exists !\gamma &\qquad \text{(1)}\\ \vdash .*34·1.\supset \vdash \colon\ldotp \text{Hp}.\supset :&M\{(Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }\}\mu . \equiv .\\ &(\exists Q,\alpha ).M = (Y\parallel \breve{X} )ʻQ.QA\alpha .\mu = S_{\times }ʻ\alpha .\\ [*114·601·602] &\equiv .(\exists \alpha ).M = (Y\parallel \breve{X} )ʻ(Sʻ\alpha )\downarrow \alpha .\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ [*113·123] &\equiv .(\exists \alpha ).M = (YʻSʻ\alpha )\downarrow (Xʻ\alpha ).\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ [*114·611] &\equiv .(\exists \alpha ).M = PʻS_{\times }ʻ\alpha .\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ [*13·193.*114·601.*71·16] &\equiv .M = Pʻ\mu .\exists !\gamma .\\ [*71·36.*80·14.(1)] &\equiv .MP\mu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*114·613. \[\begin{aligned}\vdash :\text{Hp}*114·61.&\text{Hp}*114·604.\supset .\\ &P = Tʻ(Y\downarrow X).(Y\downarrow X)\in {\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*21·33.*114·604.\supset \vdash \colon\ldotp \text{Hp}*114·604.\supset :\\ X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma .\supset .Tʻ(Y\downarrow X) = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times } &\qquad \text{(1)}\\ \vdash .(1).*114·61·612.*113·106.\supset \vdash .\text{Prop} \end{array} \]

*114·614. \(\vdash :\text{Hp}*114·604.\supset .{\in}_{\Delta}ʻ\text{D}ʻS_{\times }\subset \text{D}ʻT\)

Dem. \[ \begin{array}{l} \vdash .*114·613.\supset \vdash \colon\ldotp \text{Hp}.&\supset :P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times }.\supset .(\exists Q).P = TʻQ.\\ [*33·43] &\supset .P\in \text{D}ʻT\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*114·62. \[\begin{aligned}&\vdash :\text{Hp}*114·604.\supset .T\in 1\rightarrow 1.\text{D}ʻT = {\in}_{\Delta}ʻ\text{D}ʻS_{\times }.\text{ᗡ}ʻT = {\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \\ &[*114·604·605·614]\end{aligned}\]

*114·621. \[\begin{aligned}\vdash :S\upharpoonright \gamma \in 1\rightarrow 1.&\gamma \subset \text{ᗡ}ʻS.\Lambda {\sim}\in \gamma .\supset .\\ &{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \\ &[*114·62·601]\end{aligned}\]

The hypothesis \(\Lambda {\sim}\in \gamma\) is not necessary, since, when \(\Lambda \in \gamma\), \[ {\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\,\,\text{and}\,\,{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \]

are both \(\Lambda\). This is proved in *114·63.

*114·63. \[\begin{aligned}\vdash :S\upharpoonright \gamma \in 1\rightarrow& 1.\gamma \subset \text{ᗡ}ʻS.\supset .\\ &{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma\end{aligned}\]

Dem. \[ \begin{array}{l} \vdash .*10·24.*83·11.\supset \\ \vdash :\text{Hp}.\Lambda \in \gamma .&\supset .\Lambda \times Sʻ\Lambda \in \hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}.{\in}_{\Delta}ʻ\gamma = \Lambda .\\ [*113·114] &\supset .\Lambda \in \hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}.{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma = \Lambda .\\ [*83·11] &\supset .{\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\} = \Lambda .{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma = \Lambda &\qquad \text{(1)}\\ \vdash .(1).*73·47.*114·621.\supset \vdash .\text{Prop} \end{array} \]

[Pg 134]

The above is one of the two variants of the associative law for \({\in}_{\Delta}\) and \(\times\).

*114·631. \(\vdash .{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times \alpha \}\text{ sm }{\in}_{\Delta}ʻ\alpha \times {\in}_{\Delta}ʻ\alpha \quad\left[*114·63 \frac{I}{S}\right]\)

*114·632. \[\begin{aligned}\vdash :S\upharpoonright &\gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\gamma \cap Sʻʻ\gamma = \Lambda .\supset .\\ &{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma ) \quad[*114·63·301]\end{aligned}\]

This is the second variant of the associative law for \({\in}_{\Delta}\) and \(\times\).

*114·64. \[\begin{align}\vdash :(Rʻʻ\gamma )&\upharpoonleft R,S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻR.\gamma \subset \text{ᗡ}ʻS.\supset .\\ &{\in}_{\Delta}ʻRʻʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻz\}\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *114·63 \frac{S\mid\, \breve{R},\,Rʻʻ\gamma}{S,\,\gamma}. \supset \\ \vdash :S\mid \breve{R} \upharpoonright &Rʻʻ\gamma \in 1\rightarrow 1.Rʻʻ\gamma \subset \text{ᗡ}ʻ(S\mid \breve{R} ).\supset .\\ &{\in}_{\Delta}ʻRʻʻ\gamma \times {\in}_{\Delta}ʻSʻʻRʻʻRʻʻ\gamma \text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in Rʻʻ\gamma .\mu = \alpha \times (S\mid \breve{R} )ʻ\alpha\} &\qquad \text{(1)}\\ \vdash .*74·14 .*35·354. \supset \vdash :\text{Hp}.&\supset .S\mid \breve{R} \upharpoonright Rʻʻ\gamma = S\upharpoonright \gamma \mid \gamma \upharpoonleft \breve{R} .\breve{R} \upharpoonright Rʻʻ\gamma = \gamma \upharpoonleft \breve{R} .\\ [*71·252] &\supset .S\mid \breve{R} \upharpoonright Rʻʻ\gamma \in 1\rightarrow 1 &\qquad \text{(2)}\\ \vdash .*37·2.\supset \vdash :\text{Hp}.&\supset .Rʻʻ\gamma \subset Rʻʻ\text{ᗡ}ʻS.\\ [*37·32] &\supset .Rʻʻ\gamma \subset \text{ᗡ}ʻ(S\mid \breve{R} ) &\qquad \text{(3)}\\ \vdash .*74·171. &\supset \vdash :\text{Hp}.\supset .\breve{R} ʻʻRʻʻ\gamma = \gamma &\qquad \text{(4)}\\ \vdash .(4).*74·14. \supset \vdash :\text{Hp}.&\supset .(Rʻʻ\gamma )\upharpoonleft R = R\upharpoonright \gamma .\\ [*35·7.*71·4] &\supset .\hat{\mu} \{(\exists \alpha ).\alpha \in Rʻʻ\gamma .\mu = \alpha \times (S\mid \breve{R} )ʻ\alpha \}\\ &= \hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻ\breve{R} ʻRʻz\}\\ [*74·53] &= \hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻz\} &\qquad \text{(5)}\\ \vdash .(1).(2).(3).(4).(5).\supset \vdash .\text{Prop} \end{array} \]

In the above proposition, the hypothesis has to be such as to yield \(\breve{R} ʻʻRʻʻ\gamma = \gamma\). Various other forms of hypothesis will secure this result, and will give other forms of the above proposition. This subject is treated in *74, above.

*114·65. \[\begin{align}\vdash \colon\ldotp &(Rʻʻ\gamma )\upharpoonleft R,S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻR.\gamma \subset \text{ᗡ}ʻS.Rʻʻ\gamma \cap Sʻʻ\gamma = \Lambda .\supset .\\ &{\in}_{\Delta}ʻ(Rʻʻ\gamma \cup Sʻʻ\gamma )\text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻz\}\\ &[*114·64·301]\end{align}\]

Summary of *115.

Whenever \(\kappa\) is a class of mutually exclusive classes, \({\in}_{\Delta}ʻ\kappa\) is similar to \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\); hence \[ \Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa . \]

Now \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is of the same type as \(\kappa\); and when \(\kappa\) is a class of mutually exclusive classes, \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) consists of all classes formed by selecting one representative from each member of \(\kappa\). It often happens that \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is easier to deal with than \({\in}_{\Delta}ʻ\kappa\); hence when possible (i.e. when \(\kappa \in \text{Cls}^{2} \text{excl})\), it is convenient to use \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\), rather than \({\in}_{\Delta}ʻ\kappa\), as the standard member of \(\Pi\text{Nc}ʻ\kappa\). We therefore put \[ \text{Prod}ʻ\kappa = \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}. \]

We shall call \(\text{Prod}ʻ\kappa\) the "multiplicative class" of \(\kappa\).

The associative law, \[ \text{Prod}ʻsʻ\kappa \text{ sm } \text{Prod}ʻ\text{Prod}ʻʻ\kappa , \] requires not merely \(\kappa \in \text{Cls}^{2} \text{excl}\), but also \(sʻ\kappa \in \text{Cls}^{2} \text{excl}\). The combination of these two hypotheses gives a completely disjointed class of classes of classes, i.e. a class of classes of classes \(\kappa\) which can be obtained by dividing a given class \((sʻsʻ\kappa)\) into mutually exclusive portions, and then dividing each of those portions into mutually exclusive portions. For example, take a square (a class of points) and divide it by horizontal lines, and then divide each of the resulting rectangles by vertical lines; then the resulting rows of little rectangles form such a class, each row of rectangles being one member of the class. Such a class we call an "arithmetical" class, and denote by "\(\text{Cls}^{3}\,\text{arithm}\)."

The present number is concerned with the properties of multiplicative classes and arithmetical classes. Some of these properties will be useful in dealing with exponentiation.

The present number begins with various propositions concerning \(\text{Prod}ʻ\kappa\) which are merely repetitions of previous propositions of *83, *84, *85 or *113. Thus we have

*115·141. \(\vdash :\exists !\text{Prod}ʻ\kappa .\supset .sʻ\text{Prod}ʻ\kappa = sʻ\kappa \quad\text{by *83·66}\),

*115·142. \(\vdash .\text{Prod}ʻ\iota ʻ\alpha = \iotaʻʻ\alpha \quad\text{by *83·7}\),

*115·143. \(\vdash .\text{Prod}ʻ\iotaʻʻ\alpha = \iota ʻ\alpha \quad\text{by *83·71}\),

*115·16. \(\vdash .\kappa \in \text{Cls}^{2} \text{excl}.\supset .\text{Prod}ʻ\kappa \subset \text{Nc}ʻ\kappa \quad\text{by *100·64}\),

and various other properties.

[Pg 136]

We then proceed to consider \(\text{Cls}^{3}\,\text{arithm}\). We prove

*115·22. \(\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset :sʻʻ\kappa \in \text{Cls}^{2} \text{excl}:\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha =\beta\)

and *115·23 gives a similar proposition substituting "\(\text{Prod}\)" for \(s\).

After a few more propositions on \(\text{Cls}^{3}\,\text{arithm}\), we proceed to the associative law for \(\text{Prod}\) (*115·34), i.e. \[ \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}ʻʻ\kappa \text{ sm }\text{Prod}ʻsʻ\kappa . \]

(This proposition, *115·34, also states that, with the same hypothesis, \(\text{Prod}ʻsʻ\kappa \text{ sm }{\in}_{\Delta}ʻsʻ\kappa)\). Hence we have

*115·35. \[\begin{align}\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Nc}ʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa &=\text{Nc}ʻ\text{Prod}ʻsʻ\kappa =\Pi \text{Nc}ʻ\text{Prod}ʻʻ\kappa \\ &=\Pi \text{Nc}ʻ{\in}_{\Delta}ʻʻ\kappa =\Pi \text{Nc}ʻsʻ\kappa\end{align}\]

We have also

*115·42. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}ʻʻ\kappa =\text{D}ʻʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa =\text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\)

*115·44. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻsʻ\kappa =sʻʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa\)

We have next to prove that if two classes of classes have double similarity, so have their multiplicative classes. The proof is simple, since the double correlator is the same as for the original classes, i.e.

*115·502. \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\)

whence

*115·51. \(\vdash :\kappa \text{ sm }\text{ sm }\lambda .\supset .\text{Prod}ʻ\kappa \text{ sm }\text{ sm }\text{Prod}ʻ\lambda\)

The number ends with some propositions which result from *114·64·65 and are analogous to them. One of these is used in the following number, in proving \(\mu ^{\varpi }\times _{c}\nu ^{\varpi }=(\mu \times _{c}\nu)^{\varpi }\), namely,

*115·6. \[\begin{align}\vdash :(Rʻʻ\gamma )&\upharpoonleft R,S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻR.\gamma \subset \text{ᗡ}ʻS.Rʻʻ\gamma ,Sʻʻ\gamma \in \text{Cls}^{2} \text{excl}.\supset .\\ &\text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu =Rʻz\times Sʻz\}\end{align}\]

The subject of this number will be useful in dealing with exponentiation, since we shall define \(\mu ^\nu\) by means of \(\text{Prod}ʻ\alpha\downarrow_{,,}ʻʻ\beta\), where \(\mu=\text{N}_{0}\text{c}ʻ\alpha\) and \(\nu =\text{N}_{0}\text{c}ʻ\beta\).

*115·01. \(\text{Prod}ʻ\kappa =\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}\)

*115·02. \(\text{Cls}^{3}\,\text{arithm}=\hat{\kappa} (\kappa ,sʻ\kappa \in \text{Cls}^{2} \text{excl}) \quad\text{Df}\)

*115·1. \(\vdash .\text{Prod}ʻ\kappa =\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad[(*115·01)]\)

*115·101. \(\vdash \colon\ldotp \alpha \in \kappa .\supset _{\alpha }.\varpi \cap \alpha \in 1:\varpi \subset sʻ\kappa :\supset .\varpi \in \text{Prod}ʻ\kappa \quad[*84·411]\)

*115·11. \[\begin{align}&\vdash \colon\colon \kappa \in \text{Cls}^{2} \text{excl}.\supset \colon\ldotp \varpi \in \text{Prod}ʻ\kappa .\equiv :\alpha \in \kappa .\supset _{\alpha }.\varpi \cap \alpha \in 1:\varpi \subset sʻ\kappa\\ &[*84·412]\end{align}\]

[Pg 137]

Owing to this proposition, \(\text{Prod}ʻ\kappa\) can be treated without any reference to \({\in}_{\Delta}ʻ\kappa\) whenever \(\kappa\in \text{Cls}^{2} \text{excl}\).

*115·12. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \text{Prod}ʻ\kappa \in \Pi \text{Nc}ʻ\kappa . \text{Prod}ʻ\kappa \text{ sm } {\in}_{\Delta}ʻ\kappa \quad[*84·41]\)

It is this proposition that makes the notation \(\text{Prod}ʻ\kappa\) appropriate for the multiplicative class.

*115·13. \(\vdash : \alpha \cap \beta = \Lambda . \supset . \text{Prod}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \text{ sm } (\alpha \times \beta ) \quad[*113·152]\)

*115·131. \(\vdash : \alpha \neq \beta . \supset . \text{Prod}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = Cʻʻ(\alpha \times \beta ) \quad[*113·151]\)

*115·14. \[\begin{align}\vdash \colon\ldotp &\kappa \cap \lambda = \Lambda . \lor . sʻ\kappa \cap sʻ\lambda = \Lambda : \supset :\\ &\varpi \in \text{Prod}ʻ(\kappa \cup \lambda ) . \equiv . (\exists \rho , \sigma ) . \rho \in \text{Prod}ʻ\kappa . \sigma \in \text{Prod}ʻ\lambda . \varpi = \rho \cup \sigma\\ &[*83·64·641]\end{align}\]

*115·141. \(\vdash : \exists ! \text{Prod}ʻ\kappa . \supset . sʻ\text{Prod}ʻ\kappa = sʻ\kappa \quad[*83·66]\)

*115·142. \(\vdash . \text{Prod}ʻ\iota ʻ\alpha = \iotaʻʻ\alpha \quad[*83·7]\)

*115·143. \(\vdash . \text{Prod}ʻ\iotaʻʻ\alpha = \iota ʻ\alpha \quad[*83·71]\)

*115·144. \(\vdash : \kappa \subset 1 . \supset . \text{Prod}ʻ\kappa = \iota ʻsʻ\kappa \quad[*83·72]\)

*115·145. \[\begin{align}&\vdash \colon\ldotp \kappa \in \text{Cls}^{2} \text{excl} . \alpha \in \kappa . \mu \cap \alpha \in 1 . \supset : \mu - \alpha \in \text{Prod}ʻ(\kappa - \iota ʻ\alpha ) . \equiv . \mu \in \text{Prod}ʻ\kappa\\ &[*84·422]\end{align}\]

*115·15. \[\begin{align}&\vdash \colon\ldotp \kappa , \lambda \in \text{Cls}^{2} \text{excl} . sʻ\kappa = sʻ\lambda . \supset : \kappa \subset \text{Prod}ʻ\lambda . \equiv . \lambda \subset \text{Prod}ʻ\kappa \\ &[*84·43]\end{align}\]

*115·151. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻsʻ\kappa = \dot{s}ʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa \quad[*85·28]\)

*115·152. \(\vdash . P_{\Delta }ʻ\alpha \text{ sm } \text{Prod}ʻP \unicode{x21A7}ʻʻ\alpha \quad[*85·55]\)

*115·153. \(\vdash . {\in}_{\Delta}ʻ\kappa \text{ sm } \text{Prod}ʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*115·152]\)

*115·154. \(\vdash . \text{Prod}ʻ\in \unicode{x21A7}ʻʻ\kappa \in \Pi \text{Nc}ʻ\kappa \quad[*115·153]\)

*115·16. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \text{Prod}ʻ\kappa \subset \text{Nc}ʻ\kappa \quad[*100·64]\)

The following proposition is used in the theory of well-ordered series (*250·5).

*115·17. \(\vdash : \exists ! {\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha . \supset . \text{Prod}ʻ\text{Cl ex}ʻ\alpha = \iota ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash . *80·14 . *115·1 . *37·45 . &\supset \vdash : \text{Hp} . \supset . \exists ! \text{Prod}ʻ\text{Cl ex}ʻ\alpha &\qquad \text{(1)}\\ \vdash . *60·61 . \text{Fact} . \supset \\ \vdash \colon\ldotp R \in 1 \rightarrow \text{Cls}. R \,\unicode{x2abd}\, \in . \text{ᗡ}ʻR = \text{Cl ex}ʻ\alpha . &\supset : R \in 1 \rightarrow \text{Cls} . R \,\unicode{x2abd}\, \in . \iotaʻʻ\alpha \subset \text{ᗡ}ʻR :\\ [*51·15] &\supset : x \in \alpha . \supset _{x} . xR(\iota ʻx) :\\ [*33·14] &\supset : \alpha \subset \text{D}ʻR &\qquad \text{(2)}\\ \vdash . *83·21 . \supset \vdash : \text{Hp} (2) . &\supset . \text{D}ʻR \subset sʻ\text{Cl ex}ʻ\alpha .\\ [*60·501] &\supset . \text{D}ʻR \subset \alpha &\qquad \text{(3)}\\ \vdash . (2) . (3) . &\supset \vdash : R \in 1 \rightarrow \text{Cls}. R \,\unicode{x2abd}\, \in . \text{ᗡ}ʻR = \text{Cl ex}ʻ\alpha . \supset . \text{D}ʻR = \alpha &\qquad \text{(4)}\\ \vdash . (4) . *115·1 . *80·14 . &\supset \vdash . \text{Prod}ʻ\text{Cl ex}ʻ\alpha \subset \iota ʻ\alpha &\qquad \text{(5)}\\ \vdash . (1) . (5) . *51·4 . \supset \vdash . \text{Prop} \end{array} \]

[Pg 138]

*115·18. \(\vdash .tʻ\text{Prod}ʻ\kappa = tʻ\kappa \quad[*83·81]\)

*115·2. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}. \equiv .\kappa ,sʻ\kappa \in \text{Cls}^{2} \text{excl} \quad[(*115·02)]\)

*115·21. \[\begin{align}\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}. \equiv :&\alpha ,\beta \in \kappa .\exists !\alpha \cap \beta .\supset _{\alpha ,\beta }.\alpha = \beta :\\ &\alpha ,\beta \in \kappa .\rho \in \alpha .\sigma \in \beta .\exists !\rho \cap \sigma .\supset _{\alpha ,\beta ,\rho ,\sigma }.\rho = \sigma \\ [*115·2.*84·11]\end{align}\]

*115·211. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha ,\beta \in \kappa .\rho \in \alpha .\sigma \in \beta .\exists !\rho \cap \sigma .\supset .\alpha = \beta\)

Dem. \[ \begin{array}{l} \vdash .*115·21 .\supset \vdash :\text{Hp}.&\supset .\rho = \sigma .\rho \in \alpha .\sigma \in \beta .\\ [*13·13] &\supset .\rho \in \alpha \cap \beta .\\ [*115·21] &\supset .\alpha = \beta :\supset \vdash .\text{Prop} \end{array} \]

*115·22. \(\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset :sʻʻ\kappa \in \text{Cls}^{2} \text{excl}:\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha = \beta\)

Dem. \[ \begin{array}{l} \vdash .*40·11. \supset \vdash :\exists !sʻ\alpha \cap sʻ\beta . &\equiv .(\exists x,\rho ,\sigma ).\rho \in \alpha .\sigma \in \beta .x\in \rho .x\in \sigma .\\ [*10·35] &\equiv .(\exists \rho ,\sigma ).\rho \in \alpha .\sigma \in \beta .\exists !\rho \cap \sigma &\qquad \text{(1)}\\ \vdash .(1).*115·211.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset .\alpha = \beta . &\qquad \text{(2)}\\ [*30·37] &\supset .sʻ\alpha = sʻ\beta &\qquad \text{(3)}\\ \vdash .(2).(3).*84·11.\supset \vdash .\text{Prop} \end{array} \]

Observe that, although "\(sʻʻ\kappa \in \text{Cls}^{2} \text{excl}\)" follows from \[ \unicode{x201c}\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha = \beta ,\unicode{x201d} \] the converse implication does not hold. If there were two different classes \(\alpha\) and \(\beta\) having the same sum, we might have \(\exists !sʻ\alpha \cap sʻ\beta\), i.e. \(\exists !sʻ\alpha\), without having \(\alpha = \beta\), in spite of "\(sʻʻ\kappa \in\text{Cls}^{2} \text{excl}\)." In proofs, less use can be made of "\(sʻʻ\kappa \in \text{Cls}^{2} \text{excl}\)" than of "\(\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta.\supset_{\alpha,\beta }.\alpha = \beta\)." If \(\Lambda {\sim} \in\kappa\) or \(\iota ʻ\Lambda {\sim} \in \kappa\), the latter implies \(s\upharpoonright \kappa \in 1\rightarrow 1\).

*115·23. \[\begin{align}\vdash \colon\ldotp \kappa \in &\text{Cls}^{3}\,\text{arithm}.\supset :\\ &\text{Prod}ʻʻ\kappa \in \text{Cls}^{2} \text{excl}:\alpha ,\beta \in \kappa .\exists !\text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset _{\alpha ,\beta }.\alpha = \beta\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*83·62. &\supset \vdash :\varpi \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset .\varpi \subset sʻ\alpha \cap sʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*24·58. &\supset \vdash :\varpi \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\exists !\varpi .\supset .\exists !sʻ\alpha \cap sʻ\beta &\qquad \text{(2)}\\ \vdash .(2).*115·22. &\supset \vdash :\text{Hp}.\alpha ,\beta \in \kappa .\varpi \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\exists !\varpi .\supset .\alpha = \beta &\qquad \text{(3)}\\ \vdash .*83·16. \text{Transp}.&\supset \vdash :\Lambda \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset .\alpha = \Lambda .\beta = \Lambda &\qquad \text{(4)}\\ \vdash .(3).(4). \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\beta \in \kappa .\exists !\text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset .\alpha = \beta . &\qquad \text{(5)}\\ [*30·37] &\supset .\text{Prod}ʻ\alpha = \text{Prod}ʻ\beta &\qquad \text{(6)}\\ \vdash .(5).(6).*84·11.\supset \vdash .\text{Prop} \end{array} \]

*115·24. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}. \equiv .\in \upharpoonright \kappa ,\in \upharpoonright sʻ\kappa \in \text{Cls}\rightarrow 1 \quad[*115·2.*84·14]\)

[Pg 139]

*115·25. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .{\in}_{\Delta}ʻ\kappa \subset 1\rightarrow 1.{\in}_{\Delta}ʻsʻ\kappa \subset 1\rightarrow 1 \quad[*84·3.*115·2]\)

*115·26. \[\begin{align}\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.&\supset .\\ &{\in}_{\Delta}ʻsʻʻ\kappa \subset 1\rightarrow 1.{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \subset 1\rightarrow 1.{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \subset 1\rightarrow 1\\ &[*84·3.*115·22.*84·55.*115·23]\end{align}\]

In the above proposition, \({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\subset 1\rightarrow 1\) does not require the hypothesis \(\kappa\in \text{Cls}^{3}\,\text{arithm}\), being true always. It is merely included here for convenience of reference.

*115·27. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\kappa \subset \text{Cls}^{2} \text{excl} \quad[*115·2.*84·25.*40·13]\)

We have now to prove the associative law for "\(\text{Prod}\)," i.e. \[ \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻsʻ\kappa \text{ sm } \text{Prod}ʻ\text{Prod}ʻʻ\kappa . \]

In virtue of *115·12, we have only to prove (under the hypothesis) \[ {\in}_{\Delta}ʻsʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \] which, by *85·44, will follow from \[ {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \] which, by *114·52, will follow from \[ {\in}_{\Delta}ʻʻ\kappa \text{ sm }\text{ sm }\text{Prod}ʻʻ\kappa . \]

Now \[ \text{Prod}ʻʻ\kappa = \text{D}_{\in}ʻʻ{\in}_{\Delta}ʻʻ\kappa . \]

Thus the correlator which will give our proposition will be \(\text{D}\upharpoonright sʻ{\in}_{\Delta}ʻʻ\kappa\). We have only to prove that this is a \(1\rightarrow 1\), and the rest follows.

*115·3. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.R,S\in sʻ{\in}_{\Delta}ʻʻ\kappa .\text{D}ʻR = \text{D}ʻS.\supset .R = S\)

Dem. \[ \begin{array}{l} \vdash . *115·23. \supset \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha ,\beta \in \kappa .R\in {\in}_{\Delta}ʻ\alpha .S\in {\in}_{\Delta}ʻ\beta .\text{D}ʻR = \text{D}ʻS.\supset .\alpha = \beta &\qquad \text{(1)}\\ \vdash .*115·27.*84·4. \supset \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha \in \kappa .R,S\in {\in}_{\Delta}ʻ\alpha .\text{D}ʻR = \text{D}ʻS.\supset .R = S &\qquad \text{(2)}\\ \vdash . (1) .(2).\supset \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha ,\beta \in \kappa .R\in {\in}_{\Delta}ʻ\alpha .S\in {\in}_{\Delta}ʻ\beta .\text{D}ʻR = \text{D}ʻS.\supset .R = S &\qquad \text{(3)}\\ \vdash .(3).*10·11·23·35.*40·11.\supset \vdash .\text{Prop} \end{array} \]

*115·31. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻʻ\kappa \text{ sm }\text{ sm }{\in}_{\Delta}ʻʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*115·3.*71·55.*72·13.&\supset \vdash :\text{Hp}.\supset .\text{D}\upharpoonright sʻ{\in}_{\Delta}ʻʻ\kappa \in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash . *33·431. &\supset \vdash .sʻ{\in}_{\Delta}ʻʻ\kappa \subset \text{ᗡ}ʻ\text{D} &\qquad \text{(2)}\\ \vdash .*37·11.*115·1. &\supset \vdash .\text{Prod}ʻʻ\kappa = \text{D}_{\in}ʻʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*111·402. &\supset \vdash .\text{Prop} \end{array} \]

*115·32. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \text{ sm }{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \quad[*115·31.*114·52]\)

*115·33. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \text{ sm }{\in}_{\Delta}ʻsʻ\kappa \quad[*115·32. *85·44]\)

*115·34. \[\begin{align}&\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}ʻʻ\kappa \text{ sm }\text{Prod}ʻsʻ\kappa .\text{Prod}ʻsʻ\kappa \text{ sm }{\in}_{\Delta}ʻsʻ\kappa\\ &[*115·33·12·23]\end{align}\]

This proposition gives the associative law for "\(\text{Prod}\)."

The following proposition embodies the last three propositions.

[Pg 140]

*115·35. \[\begin{align}&\vdash : \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset .\\ &\text{Nc}ʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa = \text{Nc}ʻ\text{Prod}ʻsʻ\kappa = \Pi \text{Nc}ʻ\text{Prod}ʻʻ\kappa = \Pi \text{Nc}ʻ{\in}_{\Delta}ʻʻ\kappa = \Pi \text{Nc}ʻsʻk\\ &[*115·34·33·32]\end{align}\]

In connection with \(\text{Prod}ʻsʻ\kappa\) and \(\text{Prod}ʻ\text{Prod}ʻʻ\kappa\), there remain two propositions of sufficient interest to deserve proof, namely \[ \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻsʻ\kappa = sʻʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa \] and \[ \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻ\text{Prod}ʻʻ\kappa = \text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \]

Of these, the first is deduced from the second, while the second is proved by means of *114·51, putting \(\text{D}\) for the \(T\) which appears in that proposition, and \({\in}_{\Delta}ʻʻ\kappa\) for the \(\lambda\) of that proposition.

*115·4. \(\vdash : T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . sʻ\lambda \subset \text{ᗡ}ʻT . \supset . \text{Prod}ʻTʻʻʻ\lambda = Tʻʻʻ\text{Prod}ʻ\lambda\)

Dem. \[ \begin{array}{l} \vdash . *111·14 . *37·103 . \supset \vdash : \text{Hp} . \kappa = Tʻʻʻ\lambda . &\supset . T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\\ [*114·51 . *73·142] &\supset . {\in}_{\Delta}ʻ\kappa = (T \parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ \vdash . (1) . *115·1 . &\supset \vdash : \text{Hp} . \supset . \text{Prod}ʻTʻʻʻ\lambda = \text{D}ʻʻ(T \parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ \vdash . *37·321·231 . \supset \vdash . \text{D}ʻ(T \mid R \mid \breve{T} _{\in }) &= \text{D}ʻ(T \mid R)\\ [*37·32] &= Tʻʻ\text{D}ʻR &\qquad \text{(3)}\\ \vdash . (3) . *43·112 . \supset \vdash . \text{D}ʻʻ(T \parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &= Tʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\lambda \\ [*115·1] &= Tʻʻʻ\text{Prod}ʻ\lambda &\qquad \text{(4)}\\ \vdash . (2) . (4) . \supset \vdash . \text{Prop} \end{array} \]

*115·41. \[\begin{align}&\vdash \colon\ldotp R, S \in sʻ\lambda . \text{D}ʻR = \text{D}ʻS . \supset _{R, S} . R = S : \supset . \text{Prod}ʻ\text{D}ʻʻʻ\lambda = \text{D}ʻʻʻ\text{Prod}ʻ\lambda \\ &\left[*115·4 \frac{\text{D}}{T}. *71·55 . *72·13\right]\end{align}\]

*115·42. \[\begin{align}\vdash : \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻ\text{Prod}ʻʻ\kappa &= \text{D}ʻʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa\\ &= \text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *115·1 . &\supset \vdash . \text{Prod}ʻ\text{Prod}ʻʻ\kappa = \text{Prod}ʻ\text{D}ʻʻʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(1)}\\ \vdash . *115·3·41 . \supset \vdash : \text{Hp} . \supset . \text{Prod}ʻ\text{D}ʻʻʻ{\in}_{\Delta}ʻʻ\kappa & = \text{D}ʻʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(2)}\\ [*115·1] &= \text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(3)}\\ \vdash . (1) . (2) . (3) . \supset \vdash . \text{Prop} \end{array} \]

*115·43. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \text{Prod}ʻsʻ\kappa = sʻʻ\text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash . *115·1 . *85·28 . \supset \\ \vdash : \text{Hp} . \supset . \text{Prod}ʻsʻ\kappa &= \text{D}ʻʻ\dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \\ [*41·43] &= sʻʻ\text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa : \supset \vdash . \text{Prop} \end{array} \]

*115·44. \(\vdash : \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻsʻ\kappa = sʻʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa \quad[*115·43·42]\)

The following proposition is a lemma for *115·46.

[Pg 141]

*115·45. \[\begin{align}\vdash \colon\ldotp \alpha ,\beta \in \kappa .\exists !&sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha =\beta :\supset .\\ &(s\mid \text{D})\upharpoonright {\in}_{\Delta}ʻ\kappa \in 1\rightarrow 1.s\upharpoonright \text{Prod}ʻ\kappa \in 1\rightarrow 1\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*83·2.*40·13.&\supset \vdash :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .Rʻ\alpha \subset sʻ\alpha &\qquad \text{(1)}\\ \vdash .*83·2.*33·43.&\supset \vdash :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .Rʻ\alpha \subset sʻ\text{D}ʻR &\qquad \text{(2)}\\ \vdash .*83·23.\supset \\ \vdash :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .x\in (sʻ\text{D}ʻR\cap sʻ\alpha).&\supset .(\exists \beta ).\beta \in \kappa .x\in Rʻ\beta .x\in sʻ\alpha .\\ [(1)] &\supset .(\exists \beta ).\beta \in \kappa .x\in Rʻ\beta .x\in sʻ\beta . x\in sʻ\alpha &\qquad \text{(3)}\\ \vdash .(3).\supset \vdash \colon\ldotp \text{Hp}.R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset :x\in (sʻ\text{D}ʻR\cap sʻ\alpha ).&\supset .(\exists \beta ).x\in Rʻ\beta .\beta =\alpha . [*13·195] \supset .x\in Rʻ\alpha &\qquad \text{(4)}\\ \vdash .(1).(2).(4).&\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .Rʻ\alpha =sʻ\text{D}ʻR\cap sʻ\alpha &\qquad \text{(5)}\\ \vdash .(5).\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp R,S\in {\in}_{\Delta}ʻ\kappa .sʻ\text{D}ʻR=sʻ\text{D}ʻS.\supset :\alpha \in \kappa .\supset _{\alpha }.Rʻ\alpha =Sʻ\alpha :\\ [*33·45.*80·14] &\supset :R=S\colon\ldotp &\qquad \text{(6)}\\ [*71·55.*72·13·161]&\supset \colon\ldotp (s\mid \text{D})\upharpoonright {\in}_{\Delta}ʻ\kappa \in 1\rightarrow 1 &\qquad \text{(7)}\\ \vdash .(6).*37·63.*115·1.*30·37.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\mu ,\nu \in \text{Prod}ʻ\kappa .sʻ\mu = sʻ\nu .\supset .\mu =\nu :\\ [*71·55.*72·161] &\supset :s\upharpoonright \text{Prod}ʻ\kappa \in 1\rightarrow 1 &\qquad \text{(8)}\\ \vdash .(7).(8).\supset \vdash .\text{Prop} \end{array} \]

*115·46. \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .s\upharpoonright \text{Prod}ʻ\text{Prod}ʻʻ\kappa \in 1\rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash .*115·141.\supset \\ \vdash :\alpha ,\beta \in \kappa .\exists !sʻ\text{Prod}ʻ\alpha \cap sʻ\text{Prod}ʻ\beta .\supset .\exists !sʻ\alpha \cap sʻ\beta &&\qquad \text{(1)}\\ \vdash .(1).*115·22.\supset \\ \vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset :\alpha ,\beta \in \kappa .\exists !sʻ\text{Prod}ʻ\alpha \cap sʻ\text{Prod}ʻ\beta .&\supset .\alpha =\beta .\\ [*30·37] &\supset .\text{Prod}ʻ\alpha =\text{Prod}ʻ\beta :\\ [*37·63] \supset :\mu ,\nu \in \text{Prod}ʻʻ\kappa .\exists !sʻ\mu \cap sʻ\nu .\supset . \mu =\nu :\\ [*115·45] \supset :s\upharpoonright \text{Prod}ʻ\text{Prod}ʻʻ\kappa \in 1\rightarrow 1\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

The above proposition is used in dealing with products in relation-arithmetic (*174·42).

*115·5. \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .\text{Prod}ʻ\kappa =T_{\in }ʻʻ\text{Prod}ʻ\lambda \quad[*115·4.*111·14]\)

*115·501. \[\begin{align}&\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\exists !\text{Prod}ʻ\lambda .\supset .T\upharpoonright sʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\\ &[*115·5·141.*111·14]\end{align}\]

*115·502. \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\)

Dem. \[ \begin{array}{l} \vdash .*35·75.\supset \vdash :{\sim}\exists !\text{Prod}ʻ\lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda =\dot{\Lambda} &\qquad \text{(1)}\\ \vdash .*115·5.*37·29.\supset \vdash :\text{Hp}.{\sim}\exists !\text{Prod}ʻ\lambda .\supset .\text{Prod}ʻ\kappa =\Lambda .\\ [*37·29.*40·21] \supset .sʻ\text{Prod}ʻ\kappa =(T\upharpoonright sʻ\text{Prod}ʻ\lambda )ʻʻsʻ\text{Prod}ʻ\lambda &\qquad \text{(2)}\\ \vdash .(1).*72·1.(2).*115·5.*111·1.\supset \\ \qquad\qquad\vdash :\text{Hp}.{\sim}\exists !\text{Prod}ʻ\lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda ) &\qquad \text{(3)}\\ \vdash .(3).*115·501·141.\supset \vdash .\text{Prop} \end{array} \]

[Pg 142]

*115·51. \(\vdash : \kappa \text{ sm } \text{ sm } \lambda . \supset . \text{Prod}ʻ\kappa \text{ sm } \text{ sm } \text{Prod}ʻ\lambda \quad[*115·502]\)

The above propositions show how, in certain respects, \(\text{Prod}ʻ\kappa\) is more convenient than \({\in}_{\Delta}ʻ\kappa\). We cannot have \({\in}_{\Delta}ʻ\kappa\text{ sm } \text{ sm } {\in}_{\Delta}ʻ\lambda\), because \({\in}_{\Delta}ʻ\kappa\) is a class of relations, not a class of classes; and the correlator of \({\in}_{\Delta}ʻ\kappa\) and \({\in}_{\Delta}ʻ\lambda\) is by no means so simple a function of the correlator of \(\kappa\) and \(\lambda\) as \(T_{\in } \upharpoonright\text{Prod}ʻ\lambda\), which correlates \(\text{Prod}ʻ\kappa\) and \(\text{Prod}ʻ\lambda\), in virtue of *115·502.

The following propositions are a continuation of those given in *114·601 ff.

*115·6. \[\begin{align}\vdash : &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma , Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ &\text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻ\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *115·12 . *113·13 . \supset \\ \vdash : \text{Hp} . \supset . \text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻRʻʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma &\qquad \text{(1)}\\ \vdash . (1) . *114·64 . \supset \vdash . \text{Prop} \end{array} \]

*115·601. \[\begin{align}\vdash : (Rʻʻ\gamma ) &\upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ &\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\} \in \text{Cls}^{2} \text{excl}\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *113·19 . \supset \vdash \colon\ldotp \text{Hp} . \supset :\\ z, w \in \gamma . \exists ! (Rʻz \times Sʻz) \cap (Rʻw \times Sʻw) . &\supset . \exists ! Rʻz \cap Rʻw .\\ [*84·11] &\supset . Rʻz = Rʻw .\\ [*74·53 . *30·37] &\supset . z = w .\\ [*30·37] &\supset . Rʻz \times Sʻz = Rʻw \times Sʻw &\qquad \text{(1)}\\ \vdash . (1) . *84·11 . \supset \vdash . \text{Prop} \end{array} \]

*115·602. \[\begin{align}\vdash : &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ &\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\} \in \text{Cls}^{2} \text{excl}\\ &[\text{Proof as in *115·601}]\end{align}\]

*115·61. \[\begin{align}\vdash \colon\ldotp &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma \cap Sʻʻ\gamma = \Lambda :\\ &Rʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \lor . Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} : \supset .\\ &{\in}_{\Delta}ʻ(Rʻʻ\gamma \cup Sʻʻ\gamma ) \text{ sm } \text{Prod}ʻ\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\\ &[*115·601·602·12 . *114·65]\end{align}\]

*115·62. \[\begin{align}\vdash : &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma \cap Sʻʻ\gamma = \Lambda .\\ &(Rʻʻ\gamma \cup Sʻʻ\gamma ) \in \text{Cls}^{2} \text{excl} . \supset .\\ &\text{Prod}ʻ(Rʻʻ\gamma \cup Sʻʻ\gamma ) \text{ sm } \text{Prod}ʻ\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\\ &[*115·61·12 . *84·25]\end{align}\]

*115·63. \[\begin{align}&\vdash : (Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma , Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ &\text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm } \text{Prod}ʻ\hat{\mu}\{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\\ &[*115·6·601·12]\end{align} \]

Summary of *116.

In this number, we define "\(\alpha\, \,\text{exp}\,\, \beta\)," meaning "\(\alpha\) to the exponent \(\beta\)," where \(\alpha\) and \(\beta\) are classes, as \[ \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta . \] Now \(\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta\) consists of all ways of selecting one each from the members of \(\alpha\downarrow_{,,}ʻʻ\beta\), i.e. from the classes \(\downarrow yʻʻ\alpha\), where \(y\in \beta\). Thus to get a member of \(\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta\), take a set of couples \(x\downarrow y\), where \(x\) is always an \(\alpha\), and there is only one \(x\) for a given \(y\), and \(y\) is each member of \(\beta\) in succession. Thus for each member of \(\beta\), we have \(\text{Nc}ʻ\alpha\) possible referents; hence it is plain that the number of possible sets of couples consists of \(\text{Nc}ʻ\beta\) factors each equal to \(\text{Nc}ʻ\alpha\), and is therefore fit to be taken as defining \((\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta}\).

The definitions of \(\mu ^\nu\) and \((\text{Nc}ʻ\alpha)^{\text{Nc}ʻ\beta}\) are derived from the definition of \(\alpha \,\,\text{exp}\,\, \beta\) exactly as the definitions of \(\mu +_{c} \nu\) and \(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\), or of \(\mu \times_{c} \nu\) and \(\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta\), were derived respectively from \(\alpha + \beta\) and \(\alpha \times\beta\).

The chief difficulty in this number lies in the proof of the three formal laws of exponentiation, namely \[ \begin{aligned} \mu ^\nu \times _{c} \mu ^{\varpi } &= \mu ^{\nu +_{c} \varpi },\\ \mu ^{\varpi } \times _{c} \nu ^{\varpi } &= (\mu \times _{c} \nu )^{\varpi },\\ \text{and}\qquad (\mu ^\nu )^{\varpi } &= \mu ^{\nu \times _{c} \varpi }. \end{aligned} \] The proofs of the second and third of these, in particular, require various lemmas; but there is no difficulty involved except the complexity of the classes and relations concerned.

The definition of \(\mu ^\nu\) is so framed as to minimize the necessity for the multiplicative axiom (see the note on **113·31 in the introduction to *113). We have

*116·36. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa = \mu ^\nu\)

that is, assuming the multiplicative axiom, the product of \(\nu\) factors each equal to \(\mu\) is \(\mu ^\nu\) (assuming \(\mu\) and \(\nu\) to be cardinals which are not null).

If we had defined \(\mu ^\nu\) as the product of \(\nu\) factors each equal to \(\mu\), we should[Pg 144] have required the multiplicative axiom for almost all propositions on \(\mu ^\nu\) but by taking the particular class \(\alpha \downarrow_{,,}ʻʻ\beta\), we avoid the multiplicative axiom except in a few propositions. Among these few is the above proposition connecting exponentiation with multiplication.

Cantor has defined \(\mu ^\nu\) by means of the class of "Belegungen," i.e. the class \[ \hat{R} (R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR = \beta ) \] which (*116·12) = \((\alpha \uparrow \beta )_{\Delta }ʻ\beta\). By *85·53 and *113·103, this class is equal to \(\dot{s}ʻʻ(\alpha\,\text{exp}\,\,\beta)\) (as is proved in *116·13), whence, since \(\dot{s} \upharpoonright \alpha \,\text{exp}\,\, \beta \in 1\rightarrow1\), it follows (*116·15) that the class of "Belegungen" is similar to \(\alpha\, \,\text{exp}\,\, \beta\). Hence our definition gives the same value of \(\mu^\nu\) as Cantor's.

The propositions of the present number begin with various simple properties of \(\alpha\, \,\text{exp}\,\, \beta\). Its existence follows from

*116·152. \(\vdash :x\in \alpha .\supset .x\downarrow ʻʻ\beta \in \alpha\, \,\text{exp}\,\, \beta\)

whence (*116·16) \(\vdash .\text{Cnv}ʻʻʻ\beta \downarrow_{,,}ʻʻ\alpha \subset \alpha\, \,\text{exp}\,\, \beta\), and

*116·18. \(\vdash \colon\ldotp \exists !\alpha .\lor.\beta = \Lambda : \equiv .\exists !\alpha\, \,\text{exp}\,\, \beta\)

We have

*116·19. \(\vdash :\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\supset .(\alpha\, \,\text{exp}\,\, \beta)\text{ sm } \text{ sm }(\gamma \, \,\text{exp}\,\, \delta \)

in virtue of *113·13 and *115·51. *116·192 shows that, if \(R\upharpoonright \gamma\) correlates \(\alpha\) with \(\gamma\), and \(S\upharpoonright \delta\) correlates \(\beta\) with \(\delta\), then \((R\parallel \breve{S})\upharpoonright (\delta \times \gamma)\) is a double correlator of \(\alpha\, \,\text{exp}\,\, \beta\) with \((\gamma\,\,\text{exp}\,\, \delta)\).

We then proceed to a set of propositions on \(\mu ^\nu\) which are analogous to *113·2 ff. on \(\mu \times _{c }\nu\). We have

*116·203. \(\vdash :\exists !\mu ^\nu .\supset .\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C}\)

*116·25. \(\vdash .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } = \text{Nc}ʻ(\gamma \, \,\text{exp}\,\, \delta)\)

and various other less useful propositions.

We then have various propositions on 0 and 1 and 2. We prove

*116·301. \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .\mu ^{0} = 1\)

*116·311. \(\vdash :\nu \in \text{NC} - \iota ʻ\Lambda - \iota ʻ0.\supset .0^\nu = 0\)

*116·321. \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .\mu ^{1} = \text{ sm }ʻʻ\mu\)

(Observe that \(\text{ sm }ʻʻ\mu\) is the same cardinal as \(\mu\), but rendered typically ambiguous.)

*116·331. \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .1^\mu = 1\)

*116·34. \(\vdash .\mu ^{2} = \mu \times _{c} \mu\)

(This proposition does not require that \(\mu\) should be a cardinal.)

[Pg 145]

After the proposition (*116·36) already quoted, on the connection of exponentiation and multiplication, we proceed to a set of propositions on the case where a number of classes are all given as similar (by assignable correlations) to a given class. In *116·411, we prove that if \(\kappa\) is a class of mutually exclusive classes, each of which is similar to a given class \(\gamma\), and if, when \(\alpha \in\kappa\), \(Mʻ\alpha\) is a correlator of \(\alpha\) and \(\gamma\), and \(T\) is the sum of \(Mʻʻ\kappa\), then \[ \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma = \text{Nc}ʻ(\kappa \,\text{exp}\,\, \gamma ) = (\text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }. \] This is a further connection of multiplication and exponentiation. (On the purport of this and following propositions, see the explanation preceding *116·4.) In *116·43, the hypothesis is somewhat modified. We still have a set \(\kappa\) of classes which are all similar to \(\gamma\), but the correlator for a given class \(\alpha\) is not given as \(Mʻ\alpha\), but is given as \(Mʻw\), where \(w\) is a member of a class \(\delta\) which is similar to \(\kappa\). Then \(\kappa =\text{D}ʻʻMʻʻ\delta\). We assume that \(M\upharpoonright \delta\) is a one-one, and that if \(Mʻw\) and \(Mʻv\) have domains which overlap, then \(w = v\). Thus \(\kappa\) is a class of mutually exclusive classes, each of which has \(\text{Nc}ʻ\gamma\) terms, while \(\kappa\) has \(\text{Nc}ʻ\delta\) terms. Then it is proved in *116·43 that \[ \text{Prod}ʻ\text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } (\gamma \,\text{exp}\,\, \delta ).\Pi \text{Nc}ʻ\text{D}ʻʻMʻʻ\delta = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta }. \] This proposition and another (*116·45) which follows from it are useful in proving the formal laws of exponentiation. The proof of these occupies the following propositions from *116·5 to *116·68. We have

*116·52. \(\vdash .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{\nu +_{c} \varpi }\)

*116·55. \(\vdash .\mu ^{\varpi } x_{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi }\)

*116·63. \(\vdash .\mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi }\)

An extension of the first of these is

*116·661. \(\vdash .\Pi \text{Nc}ʻ(\alpha \,\text{exp}\,)ʻʻ\kappa = (\text{Nc}ʻ\alpha )^{\Sigma \text{Nc}ʻ\kappa }\)

Here the number of members of \(\kappa\) need not be finite. The purport of the proposition is as follows: Let \(\beta\), \(\gamma\), \(\delta\), ... be the members of \(\kappa\); form \(\alpha\,\,\text{exp}\,\,\beta\), \(\alpha \, \,\text{exp}\,\, \gamma\), \(\alpha \,\,\text{exp}\,\, \delta\), ..., and take the product of the numbers of all these; then the resulting number is the same as if we first took the sum of the numbers of all the members of \(\kappa\), thus obtaining (say) a number \(\mu\), and raised \(\text{Nc}ʻ\alpha\) to the \(\mu\)th power.

An extension of *116·55 is given by *116·68, where we prove \[ \vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Pi \text{Nc}ʻ\,\text{exp}\,\gamma ʻʻ\kappa = (\Pi \text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }. \]

There is no analogous extension of *116·63.

We prove next Cantor's proposition (which is very useful)

*116·72. \(\vdash .\text{Nc}ʻ\text{Cl}ʻ\alpha = 2^{\text{Nc}ʻ\alpha }\)

I.e. the number of combinations of \(\mu\) things any number at a time is \(2^\mu\). (Observe that \(\mu\) need not be finite.) The remainder of the number is concerned with consequences of this proposition.

[Pg 146]

*116·01. \(\alpha \,\text{exp}\,\,\beta = \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \quad\text{Df}\)

*116·02. \(\mu ^\nu =\hat{\gamma}\{(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\gamma \text{ sm }(\alpha \,\text{exp}\,\,\beta )\} \quad\text{Df}\)

*116·03. \((\text{Nc}ʻ\alpha )^\nu = (\text{N}_{0}\text{c}ʻ\alpha )^\nu \quad\text{Df}\)

*116·04. \(\mu ^{\text{Nc}ʻ\beta } = \mu ^{\text{N}_{0}\text{c}ʻ\beta } \quad\text{Df}\)

*116·1. \[\begin{align}&\vdash :\xi \in (\alpha \,\text{exp}\,\, \beta ). \equiv .(\exists R).R\in {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta .\xi = \text{D}ʻR\\ &[*115·1.(*116·01)]\end{align}\]

*116·11. \(\vdash \colon\ldotp \xi \in (\alpha \,\text{exp}\,\, \beta ). \equiv :y\in \beta .\supset _{y}.\alpha \cap \hat{x} (x\downarrow y\in \xi )\in 1:\xi \subset \beta \times \alpha \)

Dem. \[ \begin{array}{l} \vdash .*113·111.*115·11.\supset \\ \vdash \colon\ldotp \xi \in (\alpha \,\text{exp}\,\, \beta ). &\equiv :\rho \in \alpha \downarrow_{,,}ʻʻ\beta .\supset _{\rho }.\rho \cap \xi \in 1:\xi \subset sʻ\alpha \downarrow_{,,}ʻʻ\beta :\\ [*38·2.*113·1] &\equiv :y\in \beta .\supset _{y}.\downarrow yʻʻ\alpha \cap \xi \in 1:\xi \subset \beta \times \alpha &\qquad \text{(1)}\\ \vdash .*37·6.\supset \\ \vdash :\downarrow yʻʻ\alpha \cap \xi \in 1.& \equiv .\hat{R}\{(\exists x).x\in \alpha .R = x\downarrow y.R\in \xi\}\in 1.\\ [*13·193] &\equiv .\hat{R}\{(\exists x).x\in \alpha .x\downarrow y\in \xi .R = x\downarrow y\}\in 1.\\ [*37·6] &\equiv .\downarrow yʻʻ\hat{x} (x\in \alpha .x\downarrow y\in \xi )\in 1.\\ [*73·611·44] &\equiv .\hat{x} (x\in \alpha .x\downarrow y\in \xi )\in 1 &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*116·12. \(\vdash .(\alpha \uparrow \beta )_{\Delta }ʻ\beta = \hat{R} \{R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR = \beta\}\)

Dem. \[ \begin{array}{l} \vdash .*80·14.\supset \vdash :R\in (\alpha \uparrow \beta )_{\Delta }ʻ\beta . &\equiv .R\in 1\rightarrow \text{Cls}.R\,\unicode{x2abd}\, \alpha \uparrow \beta .\text{ᗡ}ʻR = \beta .\\ [*35·83] &\equiv .R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR\subset \beta .\text{ᗡ}ʻR = \beta .\\ [*22·42] &\equiv .R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR = \beta :\supset \vdash .\text{Prop} \end{array} \]

*116·13. \(\vdash .sʻʻ(\alpha \,\text{exp}\,\, \beta ) = (\alpha\uparrow \beta )_{\Delta }ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*85·53.\supset \vdash .(\alpha \uparrow \beta )_{\Delta }ʻ\beta &= \dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ(\alpha \uparrow \beta )\unicode{x21A7}ʻʻ\beta \\ [*113·103] &= \dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta \\ [*115·1.(*116·01)] &= \dot{s}ʻʻ(\alpha \,\text{exp}\,\, \beta ).\supset \vdash .\text{Prop} \end{array} \]

\((\alpha \uparrow \beta )_{\Delta }ʻ\beta\) is the class of one-many relations whose converse domain is \(\beta\) and whose domain is contained in \(\alpha\). This is what Cantor calls the "Belegungsmenge," and is used by him as the definition of exponentiation. In virtue of *116·15, his definition gives the same results as ours.

*116·131. \(\vdash .\dot{s} \upharpoonright (\alpha \,\,\text{exp}\,\, \beta )\in \{(\alpha \uparrow \beta )_{\Delta }ʻ\beta\} \,\overline{\text{ sm }}\, (\alpha \,\,\text{exp}\,\, \beta )\)

Dem. \[ \begin{array}{l} \vdash .*84·241.*113·103.&\supset \vdash .\iotaʻʻ\beta \in \text{Cls}^{2} \text{excl} .\alpha \downarrow_{,,}ʻʻ\beta = (\alpha \uparrow \beta )_{\Delta }ʻʻ\iotaʻʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*85·42.&\supset \vdash :M,N\in {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta .\dot{s} ʻ\text{D}ʻM = \dot{s} ʻ\text{D}ʻN.\supset .M = N.\\ [*30·37] &\supset .\text{D}ʻM = \text{D}ʻN &\qquad \text{(2)}\\ \vdash .(2).*37·63.*115·1.(*116·01).&\supset \vdash :\mu ,\nu \in (\alpha \,\,\text{exp}\,\, \beta ).\dot{s} ʻ\mu = \dot{s} ʻ\nu .\supset .\mu = \nu :\\ [*71·55.*72·163] &\supset \vdash .\dot{s} \upharpoonright (\alpha \,\,\text{exp}\,\, \beta )\in 1\rightarrow 1 &\qquad \text{(3)}\\ \vdash .(3).*116·13.\supset \vdash .\text{Prop} \end{array} \]

[Pg 147]

*116·14. \(\vdash .(\alpha \,\,\text{exp}\,\, \beta )\text{ sm }{\in}_{\Delta} ʻ\alpha \downarrow_{,,}ʻʻ\beta \quad[*115·12.*113·111]\)

*116·15. \(\vdash .(\alpha \,\,\text{exp}\,\, \beta )\text{ sm }(\alpha \uparrow \beta )_{\Delta }ʻ\beta \quad[*116·131]\)

*116·151 is a lemma for *116·152.

*116·151. \( \vdash :x \in \alpha .\supset .x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta ) \in {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*113·105.*72·184.\supset \vdash :\text{Hp}.&\supset .x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta ) \in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash .*34·1.*38·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :R{x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta )} \lambda .\equiv .\\ &(\exists y).R=x\downarrow y.y \in \beta . \lambda =\alpha \downarrow_{,,}y.x \in \alpha .\\ [*38·21] &\supset .R \in \lambda &\qquad \text{(2)}\\ \vdash .*37·322·401. &\supset \vdash .\text{ᗡ}ʻ{x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta )}=\alpha \downarrow_{,,}ʻʻ\beta &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*80·14.\supset \vdash .\text{Prop} \end{array} \]

*116·152. \(\vdash :x \in \alpha .\supset .x\downarrowʻʻ\beta \in (\alpha \,\,\text{exp}\,\,\beta )\)

Dem. \[ \begin{array}{l} \vdash .*37·32.*35·65.&\supset \vdash .\text{D}ʻ{x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta )}=x\downarrowʻʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*116·151·1.&\supset \vdash .\text{Prop} \end{array} \]

*116·16. \(\vdash .\text{Cnv}ʻʻʻ\beta \downarrow_{,,}ʻʻ\alpha \subset \alpha \,\,\text{exp}\,\,\beta\)

Dem. \[ \begin{array}{l} \vdash .*116·152.*55·14.\supset \vdash :x \in \alpha .&\supset .\text{Cnv}ʻʻ\downarrow xʻʻ\beta \in (\alpha \,\,\text{exp}\,\,\beta ).\\ [*38·2] &\supset .\text{Cnv}ʻʻ\beta \downarrow_{,,}x \in (\alpha \,\,\text{exp}\,\,\beta ):\supset \vdash .\text{Prop} \end{array} \]

The above propositions are useful in establishing existence-theorems, as appears in the following propositions.

*116·17. \(\vdash :\exists !\beta \downarrow_{,,}ʻʻ\alpha .\supset .\exists !\alpha \,\,\text{exp}\,\,\beta \quad[*116·16.*37·47]\)

*116·171. \(\vdash \colon\ldotp \exists !\alpha .\lor.\beta =\Lambda :\supset .\exists !\alpha \,\,\text{exp}\,\,\beta\)

Dem. \[ \begin{array}{l} \vdash .*113·113.*83·15.*51·161.&\supset \vdash :\beta =\Lambda .\supset .\exists !\alpha \,\,\text{exp}\,\,\beta &\qquad \text{(1)}\\ \vdash .*116·152. &\supset \vdash :\exists !\alpha .\supset .\exists !\alpha \,\,\text{exp}\,\,\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*116·172. \(\vdash \colon\ldotp \exists !\alpha \,\,\text{exp}\,\,\beta .\supset :\exists !\alpha .\lor.\beta =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*83·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\Lambda {\sim}\in \alpha \downarrow_{,,}ʻʻ\beta :\\ [*113·112] &\supset :{\sim}(\alpha =\Lambda .\exists !\beta ):\\ [*24·51] &\supset :\exists !\alpha .\lor.\beta =\Lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*116·18. \(\vdash \colon\ldotp \exists !\alpha .\lor.\beta =\Lambda :\equiv .\exists !\alpha \,\,\text{exp}\,\,\beta \quad[*116·171·172]\)

*116·181. \(\vdash .\alpha \,\,\text{exp}\,\,\Lambda =\iota ʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*113·113.\supset \vdash .\alpha \,\,\text{exp}\,\,\Lambda &=\text{Prod}ʻ\Lambda \\ [*83·15.*33·241] &=\iota ʻ\Lambda .\supset \vdash .\text{Prop} \end{array} \]

[Pg 148]

*116·182. \(\vdash : \exists ! \beta . \supset . \Lambda \,\,\text{exp}\,\, \beta = \Lambda \quad[*113·112 . *83·11]\)

*116·183. \(\vdash . sʻ(\alpha \,\,\text{exp}\,\, \beta ) = \beta \times \alpha\)

Dem. \[ \begin{array}{l} \vdash . *115·141 . *116·18 . \supset \vdash \colon\ldotp \exists ! \alpha . \lor . \beta &= \Lambda : \supset . sʻ(\alpha \,\,\text{exp}\,\, \beta ) = sʻ\alpha \downarrow_{,,}ʻʻ\beta \\ [*113·1] &= \beta \times \alpha &\qquad \text{(1)}\\ \vdash . *116·182 . \supset \vdash \colon\ldotp \alpha &= \Lambda . \exists ! \beta . \supset . sʻ(\alpha \,\,\text{exp}\,\, \beta ) = \Lambda \\ [*113·114] &= \beta \times \alpha &\qquad \text{(2)}\\ \vdash . (1) . (2) . \supset \vdash . \text{Prop} \end{array} \]

*116·19. \(\vdash : \alpha \text{ sm } \gamma . \beta \text{ sm } \delta . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \text{ sm } \text{ sm } (\gamma \,\,\text{exp}\,\, \delta )\)

Dem. \[ \begin{array}{l} \vdash . *113·13 . \supset \vdash : \text{Hp} . &\supset . \alpha \downarrow_{,,}ʻʻ\beta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻ\delta .\\ [*115·51] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \text{ sm } \text{ sm } (\gamma \,\,\text{exp}\,\, \delta ) : \supset \vdash . \text{Prop} \end{array} \]

*116·191. \[\begin{align}\vdash : R \in \alpha \overline{\text{ sm }} \gamma . S \in \beta \,\overline{\text{ sm }}\, \delta . \supset . &(R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\alpha \,\,\text{exp}\,\, \beta ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\, \delta ) .\\ &(R \parallel \breve{S} )_{\in}ʻʻ(\gamma \,\,\text{exp}\,\, \delta ) = \alpha \,\,\text{exp}\,\, \beta \\ [*113·127 . *115·502 . *116·183]\end{align}\]

*116·192. \[\begin{align}\vdash : R \upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma . S \upharpoonright \delta \in \beta &\,\overline{\text{ sm }}\, \delta . \supset .\\ &(R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\alpha \,\,\text{exp}\,\, \beta ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\, \delta ) .\\ &(R \parallel \breve{S} )_{\in } \upharpoonright (\gamma \,\,\text{exp}\,\, \delta ) \in (\alpha \,\,\text{exp}\,\, \beta ) \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\, \delta )\\ &[*113·127 . *115·502 . *116·183 . *111·15]\end{align}\]

*116·194. \[\begin{align}\vdash : R \upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma . &S \upharpoonright \delta \in \beta \,\overline{\text{ sm }}\, \delta . \supset .\\ &(R \parallel \breve{S} ) \upharpoonright {(\gamma \uparrow \delta )_{\Delta }ʻ\delta } \in {(\alpha \uparrow \beta )_{\Delta }ʻ\beta } \,\overline{\text{ sm }}\, {(\gamma \uparrow \delta )_{\Delta }ʻ\delta }\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *116·12 . \supset \vdash : \text{Hp}. &\supset .sʻ\text{D}ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta \subset \gamma . sʻ\text{ᗡ}ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta \subset \delta .\\ [*74·773 . *73·142] \supset . (R \parallel \breve{S} ) &\upharpoonright {(\gamma \uparrow \delta )_{\Delta }ʻ\delta } \in \\ &\{(R \parallel \breve{S} )ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta\} \,\overline{\text{ sm }}\, \{(\gamma \uparrow \delta )_{\Delta }ʻ\delta\} &\qquad \text{(1)}\\ \vdash . *116·192 . *111·14 . \supset \vdash : \text{Hp} . &\supset . \alpha \,\,\text{exp}\,\, \beta = (R \parallel \breve{S} )_{\in}ʻʻ(\gamma \,\,\text{exp}\,\, \delta ) .\\ [*116·13] \supset . (\alpha \uparrow \beta )_{\Delta }ʻ\beta &= \dot{s}ʻʻ(R \parallel \breve{S} )_{\in}ʻʻ(\gamma \,\,\text{exp}\,\, \delta )\\ [*43·43] &= (R \parallel \breve{S} )ʻʻ\dot{s}ʻʻ(\gamma \,\,\text{exp}\,\, \delta )\\ [*116·13] &= (R \parallel \breve{S} )ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta &\qquad \text{(2)}\\ \vdash . (1) . (2) . \supset \vdash . \text{Prop} \end{array} \]

The following propositions (down to *116·27 exclusive) are the analogues of propositions with the same decimal part in *113.

*116·2. \(\vdash : \xi \in \mu ^\nu . \equiv . (\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta . \xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta ) \quad[(*116·02)]\)

*116·201. \[\begin{align}&\vdash \colon\ldotp \xi \in \mu ^\nu . \equiv : \mu , \nu \in \text{NC} : (\exists \alpha , \beta ) . \alpha \in \mu . \beta \in \nu . \xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta )\\ &[*116·2 . *103·27]\end{align}\]

[Pg 149]

*116·202. \[\begin{align}&\vdash \colon\ldotp \xi \in \mu ^\nu . \equiv :\exists !\mu .\exists !\nu :(\exists \alpha ,\beta ).\mu = \text{Nc}ʻ\alpha .\nu = \text{Nc}ʻ\beta .\xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta )\\ &[\text{Proof as in}\, *113·202]\end{align}\]

*116·203. \(\vdash :\exists !\mu ^\nu .\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\mu ,\nu \in \text{N}_0\text{C} \quad[*116·201·202·2]\)

*116·204. \(\vdash \colon\ldotp \mu = \Lambda .\lor.\nu = \Lambda .\lor.{\sim}(\mu ,\nu \in \text{NC}):\supset .\mu ^\nu = \Lambda \quad[*116·203]\)

*116·205. \(\vdash :{\sim}(\mu ,\nu \in \text{N}_0\text{C}).\supset .\mu ^\nu = \Lambda \quad [*116·203]\)

*116*21. \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu ^\nu . \equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta ) \quad[*116·201]\)

*116·22. \[\begin{align}&\vdash :\xi \in {\text{Nc}(\eta )ʻ\gamma }^{\text{Nc}(\zeta )ʻ^\delta }. \equiv .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\xi \text{ sm } (\gamma \,\,\text{exp}\,\, \delta )\\ &[\text{Proof as in *113·22, using *116·19 in place of *113·13}]\end{align}\]

*116·221. \[\begin{align}&\vdash :\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\supset .{\text{Nc}(\eta )ʻ\gamma }^{\text{Nc}(\zeta )ʻ\delta } = \text{Nc}ʻ(\gamma \,\,\text{exp}\,\, \delta )\\ &[*116·22]\end{align}\]

*116·222. \(\vdash .(\text{N}_0\text{c}ʻ\gamma )^{\text{N}_0\text{c}ʻ\delta } = \text{Nc}ʻ(\gamma \,\,\text{exp}\,\, \delta ) \quad[\text{Proof as in *113·222}]\)

*116·23. \(\vdash .\mu ^\nu \in \text{NC} \quad[\text{Proof as in *113·23}]\)

*116·24. \(\vdash .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } = (\text{N}_0\text{c}ʻ\gamma )^{\text{N}_0\text{c}ʻ\delta } \quad[(*116·03·04)]\)

*116·25. \(\vdash .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } = \text{Nc}ʻ(\gamma \,\,\text{exp}\,\, \delta ) \quad[*116·24·222]\)

*116·251. \(\vdash .(\gamma \,\,\text{exp}\,\, \delta )\in (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } \quad[*116·25.*100·3]\)

*116·26. \[\begin{align}&\vdash :\mu ,\nu \in \text{NC}.\exists !\text{ sm }_\eta ʻʻ\mu .\exists !\text{ sm }_\zetaʻʻ\nu .\supset .\mu ^\nu = (\text{ sm }_\eta ʻʻ\mu )^{\text{ sm }_\zetaʻʻ\nu }\\ &[\text{Proof as in *113·26}]\end{align}\]

This proposition shows that we may raise or lower the types of \(\mu\) and \(\nu\) as we please, without affecting the value of \(\mu ^\nu\) provided \(\mu\) and \(\nu\), or rather \(\text{ sm }ʻʻ\mu\) and \(\text{ sm }ʻʻ\nu\), exist in the new types.

*116·261. \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu ^\nu = \{\mu ^{(1)}\}^{\nu ^{(1)}} = \{\mu _{(00)}\}^{\nu _{(00)}} = etc. \quad[\text{Proof as in *113·261}]\)

Here "etc." covers any derivative of \(\mu\) or \(\nu\) whose existence follows from that of \(\mu\) or \(\nu\).

*116·27. \[\begin{align}&\vdash .\mu ^\nu = \hat{\xi} \{(\exists \alpha ,\beta ).\mu = \text{N}_0\text{c}ʻ\alpha .\nu = \text{N}_0\text{c}ʻ\beta .\xi \text{ sm } (\alpha \uparrow \beta )_{\Delta }ʻ\beta\}\\ &[*116·15.*73·37.(*116·02)]\end{align}\]

*116·271. \(\vdash :\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\supset .(\alpha \,\,\text{exp}\,\, \beta )\in \mu ^\nu \quad[*116·21]\)

*116·3. \(\vdash .(\text{Nc}ʻ\alpha )^{0} = 1\)

Dem. \[ \begin{array}{l} \vdash .*101·1.*116·25.\supset \vdash .(\text{Nc}ʻ\alpha )^{0} &= \text{Nc}ʻ(\alpha \,\,\text{exp}\,\, \Lambda )\\ [*116·181] &= \text{Nc}ʻ\iota ʻ\Lambda \\ [*101·2] & = 1.\supset \vdash .\text{Prop} \end{array} \]

*116·301. \(\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\supset .\mu ^{0} = 1 \quad[\text{Proof as in *113·601}]\)

*116·31. \(\vdash :\beta \neq \Lambda .\supset .0^{\text{Nc}ʻ\beta } = 0\)

Dem. \[ \begin{array}{l} \vdash .*101·1.*116·25.&\supset \vdash .0^{\text{Nc}ʻ\beta } = \text{Nc}ʻ(\Lambda \,\,\text{exp}\,\, \beta )\\ [*116·182] \supset \vdash :\text{Hp}.\supset .0^{\text{Nc}ʻ\beta } &= \text{Nc}ʻ\Lambda\\ [*101·1] &= 0:\supset \vdash .\text{Prop} \end{array} \]

[Pg 150]

*116·311. \(\vdash :\nu \in \text{NC}-\iota ʻ\Lambda -\iota ʻ0.\supset .0^{\nu }=0\)

Dem. \[ \begin{array}{l} \vdash .*103·34.*101·1.\supset \vdash :\text{Hp}.&\supset .(\exists \beta ).\beta \neq \Lambda .\nu =\text{N}_{0}\text{c}ʻ\beta .\\ [*13·12·15] \supset .(\exists \beta ).\beta \neq \Lambda .0^{\nu }&=0^{\text{N}_{0}\text{c}ʻ\beta }\\ [*116·31.(*116·04)] &=0:\supset \vdash .\text{Prop} \end{array} \]

*116·32. \(\vdash .(\text{Nc}ʻ\alpha )^{1}=\text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*116·25.*101·2.\supset \vdash .(\text{Nc}ʻ\alpha )^{1}&=\text{Nc}ʻ{\alpha \,\,\text{exp}\,\, (\iota ʻx)}\\ [(*116·01)] &=\text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\iota ʻx\\ [*115·142.*53·31] &=\text{Nc}ʻ\iota ʻʻ\alpha \downarrow_{,,}x\\ [*113·11.*100·6] &=\text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} \end{array} \]

*116·321. \(\vdash .\mu \in \text{NC}-\iota ʻ\Lambda .\supset .\mu ^{1}=\text{ sm }ʻʻ\mu \quad[*116·32]\)

It would not be an error to write "\(\mu ^{1}=\mu\)" instead of "\(\mu ^{1}=\text{ sm }ʻʻ\mu\)" in the above proposition. For if the "\(\text{ sm }\)" is typically determined so that \(\text{sm}ʻʻ\mu \inʻ\mu\), then \(\text{ sm }ʻʻ\mu =\mu\). Thus in virtue of *116·321, \(\mu ^{1}=\mu\) is true whenever it is significant. But the above form gives more information, since it preserves the typical ambiguity of \(\mu ^{1}\) and \(\text{ sm }ʻʻ\mu\).

*116·33. \(\vdash .1^{\text{Nc}ʻ\beta }=1\)

Dem. \[ \begin{array}{l} \vdash .*113·11. \supset \vdash :\alpha \in 1.&\supset .\alpha \downarrow_{,,}ʻʻ\beta \subset 1.\\ [*115·144.*101·2] &\supset .\text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta =1 &\qquad \text{(1)}\\ \vdash .(1).*101·2. &\supset \vdash .\text{Nc}ʻ{(\iota ʻx) \,\,\text{exp}\,\, \beta }=1 &\qquad \text{(2)}\\ \vdash .*101·2.*116·25.&\supset \vdash .1^{\text{Nc}ʻ\beta }=\text{Nc}ʻ\{(\iota ʻx) \,\,\text{exp}\,\, \beta\} &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*116·311. \(\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\supset .1^{\mu }=1\)

Dem. \[ \begin{array}{l} \vdash .*103·34.\supset \vdash :\text{Hp}.&\supset .(\exists \beta ).\mu =\text{N}_{0}\text{c}ʻ\beta .\\ [*13·12·15] &\supset .(\exists \beta ).1^{\mu }=1^{\text{N}_{0}\text{c}ʻ\beta }.\\ [(*116·04)] &\supset .(\exists \beta ).1^{\mu }=1^{\text{Nc}ʻ\beta }.\\ [*116·33] &\supset .1^{\mu }=1:\supset \vdash .\text{Prop} \end{array} \]

*116·34. \(\vdash .\mu ^{2}=\mu \times _{c}\mu\)

Dem. \[ \begin{array}{l} \vdash .*24·1.*101·3.\supset \vdash .\iota ʻ\Lambda \cup \iota ʻ\text{V}\in 2.\\ [*116·222] \supset \vdash :\mu &=\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{2}=\text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V})\\ [*53·32] &=\text{Nc}ʻ\text{Prod}ʻ(\iota ʻ\alpha \downarrow_{,,}\Lambda \cup \iota ʻ\alpha \downarrow_{,,}\text{V})\\ [*115·13.*55·233.*38·2] &=\text{Nc}ʻ(\alpha \downarrow_{,,}\Lambda \times \alpha \downarrow_{,,}\text{V})\\ [*113·11·25·13] &=\text{Nc}ʻ\alpha \times _{c}\text{Nc}ʻ\alpha \\ [*113·24] &=\mu \times _{c}\mu &\qquad \text{(1)}\\ \vdash .(1).*103·2.&\supset \vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu ^{2}=\mu \times _{c}\mu &\qquad \text{(2)}\\ \vdash .*116·205. \supset \vdash :\mu {\sim}\in \text{N}_{0}\text{C}.\supset .\mu ^{2}&=\Lambda \\ [*113·205] &=\mu \times _{c}\mu &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

[Pg 151]

*116·35. \(\vdash :\mu ^{\nu }=0.\equiv .\mu =0.\nu \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*116·311.&\supset \vdash :\mu =0.\nu \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda .\supset .\mu ^{\nu }=0 &\qquad \text{(1)}\\ \vdash .*101·12. \supset \vdash :\mu ^{\nu }=0.&\supset .\exists !\mu ^{\nu }.\\ [*116·203] &\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda &\qquad \text{(2)}\\ \vdash .(2)*116·21.*54·102.&\supset \\ \vdash \colon\ldotp \mu ^{\nu }=0.&\supset :\xi =\Lambda .\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta ):\\ [*73·47] &\supset :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\alpha \,\,\text{exp}\,\, \beta =\Lambda :\\ [*116·18] &\supset :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\alpha =\Lambda .\beta \neq \Lambda :\\ [*13·195] &\supset :\Lambda \in \mu .\nu \neq \iota ʻ\Lambda .\exists !\nu :\\ [*101·1.*100·45.(2)] &\supset :\mu =0.\nu \in \text{NC}-\iota ʻ\Lambda -\iota ʻ0 &\qquad \text{(3)}\\ \vdash .(1).(3).\supset \vdash .\text{Prop} \end{array} \]

*116·351. \[\begin{align}&\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\kappa =\Lambda .\nu =0.\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa =1\\ &[*116·301.*114·2]\end{align}\]

*116·352. \[\begin{align}&\vdash :\mu =0.\nu \in \text{NC}-\iota ʻ\Lambda .\kappa \in \nu .\Lambda \in \kappa .\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa =0\\ &[*116·311.*114·23]\end{align}\]

*116·353. \(\vdash :\mu =0.\nu \in \text{NC}-\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*60·362.*54·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\kappa =\Lambda .\lor.\kappa =\iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash .*100·45.*101·1.\supset \vdash :\text{Hp}.\kappa =\Lambda .&\supset .\nu =0.\\ [*116·351] &\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa &\qquad \text{(2)}\\ \vdash .*51·16. &\supset \vdash :\text{Hp}.\kappa =\iota ʻ\Lambda .\supset .\Lambda \in \kappa .\\ [*116·352] &\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*116·36. \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\mu ^\nu\)

Dem. \[ \begin{array}{l} \vdash .*113·12.*100·45.\supset \vdash :\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\exists !\alpha .\supset .\alpha \downarrow_{,,}ʻʻ\beta \in \nu \cap \text{Cl}ʻ\mu &\qquad \text{(1)}\\ \vdash .(1).*114·571. \supset \vdash \colon\ldotp \text{Mult ax}.\supset :\\ \mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\exists !\alpha .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\Pi \text{Nc}ʻ\alpha \downarrow_{,,}ʻʻ\beta \\ [*116·14.*114·1] =\text{Nc}ʻ(\alpha \,\,\text{exp}\,\, \beta )\\ [*116·271] =\mu ^\nu &\qquad \text{(2)}\\ \vdash .(2).\supset \vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\exists !\mu -\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\mu ^\nu &\qquad \text{(3)}\\ \vdash .*51·4.*54·1.\supset \vdash :\mu \in \text{NC}-\iota ʻ\Lambda .{\sim}\exists !\mu -\iota ʻ\Lambda .\supset .\mu =0 &\qquad \text{(4)}\\ \vdash .(4).*116·353.\supset \\ \vdash :\mu ,\nu \in \text{Nc}-\iota ʻ\Lambda .{\sim}\exists !\mu -\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\mu ^\nu &\qquad \text{(5)}\\ \vdash .(3).(5).\supset \vdash .\text{Prop} \end{array} \]

[Pg 152]

In the above proposition, "\(\nu \in \text{NC}\)" is sufficient hypothesis as to \(\nu\), since "\(\nu \neq \Lambda\)" is implied by \(\kappa \in \nu \cap \text{Cl}ʻ\mu\). But \(\mu \neq \Lambda\) is essential, since if \(\mu = \Lambda\), \(\mu ^\nu = \Lambda\) and \(\kappa= \Lambda\) (provided \(\nu = 0)\), whence \(\Pi \text{Nc}ʻ\kappa = 1\).

The above proposition connects exponentiation with multiplication.

*116·361. \(\vdash \colon\ldotp \text{Mult ax} . \supset : \mu , \nu \in \text{NC} - \iota ʻ\Lambda . \kappa \in \nu \cap \text{Cl excl}ʻ\mu . \supset . \text{Prod}ʻ\kappa \in \mu ^\nu\)

Dem. \[ \begin{array}{l} \vdash . *115·12 . \supset \vdash : \kappa \in \nu \cap \text{Cl excl}ʻ\mu . \supset . \text{Prod}ʻ\kappa \in \Pi \text{Nc}ʻ\kappa &\qquad \text{(1)}\\ \vdash . (1) . *116·36 . \supset \vdash . \text{Prop} \end{array} \]

The following propositions, which illustrate certain generalizations of the relations of rows and columns, may be made clearer by the accompanying figure, in which, for the sake of simplicity, all the classes concerned are taken to be finite.

Let \(\kappa\) be a set of classes, constituted by four rows of five dots in the figure, which are each given as similar to a given class \(\gamma\), represented by the top row of five dots in the figure, namely the row enclosed in an oval. We assume that an actual correlating relation is given correlating each member of \(\kappa\) with \(\gamma\). Let \(\lambda\) be the class of these relations, and assume that \(\lambda\) consists of one correlator for each member of \(\kappa\), and that \(\kappa \in \text{Cls}^{2} \text{excl}\). Thus \(\text{D}ʻʻ\lambda = \kappa\), and \(R \in \lambda . \supset. \text{ᗡ}ʻR = \gamma\). Put \(T = \dot{s} ʻ\lambda\). Then, if \(z\in \gamma\), \(T\) relates to \(z\) every member of the column below \(z\), i.e. \(\overrightarrow{T}ʻz\) consists of the four dots which are vertically below \(z\); assuming, what in the circumstances is possible, that each dot is placed below its correlate in \(\gamma\). Thus \(\overrightarrow{T}ʻʻ\gamma\) represents the columns, while \(\text{D}ʻʻ\lambda\) represents the rows.

We prove, in *116·41, that \(\overrightarrow{T}ʻʻ\gamma\), the class of rows, has double similarity with \(\lambda\downarrow_{,,}ʻʻ\gamma\), or, what comes to the same thing, with \(\kappa\downarrow_{,,}ʻʻ\gamma\). Hence it follows that \(Tʻʻ\gamma\), which is the whole class of dots, is similar to \(\gamma\times \lambda\) or \(\gamma \times \kappa\), and that \(\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma\), which is the product of [Pg 153]the numbers of the columns, is equal to \((\text{Nc}ʻ\lambda)^{\text{Nc}ʻ\gamma}\) or \((\text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma}\). The correlator which is used for proving these propositions is \(W\), where, if \(R\) is a member of \(\lambda\) and \(z\) is a member of \(\gamma\), \(W\) correlates \(Rʻz\) with \(R\downarrow z\).

Similarly, by correlating \(Rʻz\) with \(z\downarrow R\), calling the correlator \(U\), we have \(Uʻʻ\downarrow Rʻʻ\gamma =Rʻʻ\gamma\), i.e. \(U_{\in }ʻ\gamma \downarrow_{,,}R=\text{D}ʻR\), whence \(U_{\in }ʻʻ\gamma \downarrow_{,,}ʻʻ\lambda =\text{D}ʻʻ\lambda\). Hence \(\text{D}ʻʻ\lambda\), i.e. the class of rows, has double similarity with \(\gamma \downarrow_{,,}ʻʻ\lambda\) or \(\gamma \downarrow_{,,}ʻʻ\kappa\), whence the product of the numbers of the rows is \((\text{Nc}ʻ\gamma)^{\text{Nc}ʻ\lambda}\) or \((\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\kappa}\).

Finally, we take a class \(\delta\) similar to \(\kappa\) or \(\lambda\) (illustrated in the figure by the column of dots enclosed in an oval), and calling \(M\) a correlator of \(\lambda\) and \(\delta\), we replace \(\lambda\) by \(Mʻʻ\delta\) and \(\kappa\) by \(\text{D}ʻʻMʻʻ\delta\). We thus find that, if \(M\upharpoonright\delta\) correlates with \(\delta\) a class of relations whose domains are mutually exclusive, and which each correlate their domains with a given class \(\gamma\), then \(\text{D}ʻʻMʻʻ\delta\) has double similarity with \(\gamma \downarrow_{,,}ʻʻ\delta\), whence the same results as before with \(\delta\) in place of \(\kappa\) or \(\lambda\).

The following propositions are useful in connecting multiplication with exponentiation, and in proving the formal laws of exponentiation.

*116·4·401 are lemmas for *116·41.

*116·4. \[\begin{align}\vdash \colon\ldotp \lambda \subset 1&\rightarrow 1:R,S\in \lambda .\exists !\text{D}ʻR\cap \text{D}ʻS.\supset _{R,S}.R=S:\\ &\text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma .W=\hat{x} \hat{P} \{(\exists R,z).R\in \lambda .x=Rʻz.P=R\downarrow z\}:\\ &\supset .W\in 1\rightarrow 1.\text{ᗡ}ʻW=\gamma \times \lambda .\text{D}ʻW=\text{D}ʻ\dot{s} ʻ\lambda\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*21·33.\supset \vdash \colon\ldotp \text{Hp}.&\supset :xWP.xWQ.\equiv .\\ &(\exists R,S,z,w).R,S\in \lambda .x=Rʻz=Sʻw.P=R\downarrow z.Q=S\downarrow w.\\ [*33·43] &\equiv .(\exists R,S,z,w).R,S\in \lambda .x=Rʻz=Sʻw.x\in \text{D}ʻR\cap \text{D}ʻS.\\ &P=R\downarrow z.Q=S\downarrow w.\\ [\text{Hp}.*13·195] &\supset .(\exists R,z,w).R\in \lambda .x=Rʻz=Rʻw.P=R\downarrow z.Q=R\downarrow w.\\ [*71·532.*13·195] &\supset .(\exists R,z).R\in \lambda .x=Rʻz.P=R\downarrow z.Q=R\downarrow z.\\ [*13·172] &\supset .P=Q &\qquad \text{(1)}\\ \vdash .*21·33.\supset \vdash \colon\ldotp \text{Hp}.\supset :xWP.yWP.&\equiv .\\ &(\exists R,S,z,w).R,S\in \lambda .x=Rʻz.y=Sʻw.P=R\downarrow z=Q\downarrow w.\\ [*55·202] &\supset .(\exists R,S,z,w).R,S\in \lambda .x=Rʻz.y=Sʻw.R=S.z=w.\\ [*13·22·172] &\supset .x=y &\qquad \text{(2)}\\ \vdash .*33·131.\supset \vdash \colon\ldotp \text{Hp}.\supset :P\in \text{ᗡ}ʻW.&\equiv .(\exists x,R,z).R\in \lambda .x=Rʻz.P=R\downarrow z.\\ [*71·411] &\equiv .(\exists R,z).R\in \lambda .z\in \text{ᗡ}ʻR.P=R\downarrow z.\\ [\text{Hp}] &\equiv .(\exists R,z).R\in \lambda .z\in \gamma .P=R\downarrow z.\\ [*113·101] &\equiv .P\in \gamma \times \lambda &\qquad \text{(3)}\\ \vdash .*33·13. \supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \text{D}ʻW.&\equiv .(\exists P,R,z).R\in \lambda .x=Rʻz.P=R\downarrow z.\\ [*55·12.*71·36] &\equiv .(\exists R,z).R\in \lambda .xRz.\\ [*41·11.*33·13] &\equiv .x\in \text{D}ʻ\dot{s} ʻ\lambda &\qquad \text{(4)}\\ \vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} \end{array} \]

[Pg 154]

*116·401. \(\vdash : \text{Hp} *116·4 . T = \dot{s} ʻ\lambda . \supset . \overrightarrow{T}ʻʻ\gamma = W_{\in }ʻʻ\lambda \downarrow_{,,} ʻʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *37·11·1 . *38·2 . &\supset \vdash \colon\ldotp \text{Hp} . z \in \gamma . \supset :\\ x \in W_{\in }ʻ\lambda \downarrow_{,,} z . &\equiv . (\exists R) . R \in \lambda . x W (R \downarrow z) .\\ [*21·33] &\equiv . (\exists R, S, w) . R, S \in \lambda . x = Sʻw . R \downarrow z = S \downarrow w .\\ [*55·202 . *13·22] &\equiv . (\exists R) . R \in \lambda . x = Rʻz .\\ [*71·36] &\equiv . (\exists R) . R \in \lambda . xRz .\\ [*41·11] &\equiv . x (\dot{s} ʻ\lambda ) z .\\ [\text{Hp} . *32·18] &\equiv . x \in \overrightarrow{T}ʻz &\qquad \text{(1)}\\ \vdash . (1) . *37·68 . \supset \vdash . \text{Prop} \end{array} \]

*116·41. \[\begin{align}&\vdash \colon\ldotp \lambda \subset 1 \rightarrow 1 . \text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma : R, S \in \lambda . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R, S} . R = S : T = \dot{s} ʻ\lambda :\\ &\supset . \overrightarrow{T}ʻʻ\gamma \text{ sm } \text{ sm } \lambda \downarrow_{,,} ʻʻ\gamma . Tʻʻ\gamma \text{ sm } \gamma \times \lambda . T \in \text{Cls} \rightarrow 1 . \overrightarrow{T}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} .\\ &\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma = \text{Nc}ʻ\text{Prod}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻ(\lambda \,\,\text{exp}\,\, \gamma ) = (\text{Nc}ʻ\lambda )^{\text{Nc}ʻ\gamma }\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *116·4·401 . *111·4 . *113·1 . \supset \vdash : \text{Hp} . &\supset . \overrightarrow{T}ʻʻ\gamma \text{ sm } \text{ sm } \lambda \downarrow_{,,} ʻʻ\gamma . &\qquad \text{(1)}\\ [*111·44 . *40·5] &\supset . Tʻʻ\gamma \text{ sm } \gamma \times \lambda &\qquad \text{(2)}\\ \vdash . *72·321 . *85·14 . &\supset \vdash : \text{Hp} . \supset . T \in \text{Cls} \rightarrow 1 . \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma &\qquad \text{(3)}\\ \vdash . (3) . *84·51 . \supset \vdash : \text{Hp} . &\supset .\overrightarrow{T}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} . &\qquad \text{(4)}\\ [*115·12] &\supset . \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻ\text{Prod}ʻ\overrightarrow{T}ʻʻ\gamma &\qquad \text{(5)}\\ \vdash . (1) . *114·52 . \supset \vdash : \text{Hp} . \supset . \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma &= \text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \downarrow_{,,} ʻʻ\gamma \\ [*116·14] &= \text{Nc}ʻ(\lambda \,\,\text{exp}\,\, \gamma ) &\qquad \text{(6)}\\ [*116·25] &= (\text{Nc}ʻ\lambda )^{\text{Nc}ʻ\gamma } &\qquad \text{(7)}\\ \vdash . (1) . (2) . (3) . (4) . (5) . (6) . (7) . \supset \vdash . \text{Prop} \end{array} \]

The following proposition is merely another form of *116·41.

*116·411. \[\begin{align}&\vdash \colon\ldotp \kappa \in \text{Cls}^{2} \text{excl} : \alpha \in \kappa . \supset _{\alpha } . Mʻ\alpha \in \alpha \,\overline{\text{ sm }}\, \gamma : T = \dot{s} ʻMʻʻ\kappa ʻʻ: \supset .\\ &\overrightarrow{T}ʻʻ\gamma \text{ sm } \text{ sm } \kappa \downarrow_{,,} ʻʻ\gamma . Tʻʻ\gamma \text{ sm } \gamma \times \kappa . T \in \text{Cls} \rightarrow 1 . \overrightarrow{T}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} .\\ &\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma = \text{Nc}ʻ\text{Prod}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻ(\kappa \,\,\text{exp}\,\, \gamma ) = (\text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *73·03 . &\supset \vdash : \text{Hp} . \supset . Mʻʻ\kappa \subset 1 \rightarrow 1 . \text{ᗡ}ʻʻMʻʻ\kappa \subset \iota ʻ\gamma &\qquad \text{(1)}\\ \vdash . *111·16 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha , \beta \in \kappa . Mʻ\alpha = Mʻ\beta . \supset . \alpha = \beta &\qquad \text{(2)}\\ \vdash . *14·21 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \kappa . \supset . \text{E}! Mʻ\alpha &\qquad \text{(3)}\\ \vdash . (2) . (3) . *73·24 . &\supset \vdash : \text{Hp} . \supset . Mʻʻ\kappa \text{ sm } \kappa &\qquad \text{(4)}\\ \vdash . *73·03 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \kappa . \supset . \text{D}ʻMʻ\alpha = \alpha :\\ [*13·12] &\supset : \alpha , \beta \in \kappa . \exists ! \text{D} ʻMʻ\alpha \cap \text{D}ʻMʻ\beta . \supset . \exists ! \alpha \cap \beta .\\ [*84·11] &\supset . \alpha = \beta .\\ [*30·37 . (3)] &\supset . Mʻ\alpha = Mʻ\beta :\\ [*37·63] &\supset : R, S \in Mʻʻ\kappa . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset . R = S &\qquad \text{(5)}\\ \vdash . (1) . (4) . (5) . *116·41 \frac{Mʻʻ\kappa}{\lambda}. *113·13 . *116·19 . \supset \vdash . \text{Prop} \end{array} \]

[Pg 155]

*116·412·413 are lemmas for *116·414.

*116·412. \[\begin{align}&\vdash \colon\ldotp \lambda \subset 1 \rightarrow 1 : R, S \in \lambda . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R, S} . R = S : \text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma :\\ &U = \hat{x} \hat{P}\{(\exists R, z) . R \in \lambda . x = Rʻz . P = z \downarrow R\} :\\ &\supset . U \in (sʻ\text{D}ʻʻ\lambda ) \,\overline{\text{ sm }}\, (\lambda \times \gamma ) \quad[\text{Proof as in *116·4}]\end{align}\]

*116·413. \(\vdash : \text{Hp} *116·412 . \supset . \text{D}ʻʻ\lambda = U_{\in}ʻʻ\gamma \downarrow_{,,}ʻʻ\lambda \quad[\text{Proof as in *116·401}]\)

*116·414. \[\begin{align}&\vdash : \text{Hp} *116·412 . \supset . U \in (\text{D}ʻʻ\lambda ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,}ʻʻ\lambda ) . (\text{D}ʻʻ\lambda ) \text{ sm } \text{ sm } (\gamma \downarrow_{,,}ʻʻ\lambda )\\ &[*116·412·413]\end{align}\]

*116·42. \[\begin{align}&\vdash \colon\ldotp \lambda \subset 1 \rightarrow 1 : R, S \in \lambda . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R,S} . R = S : \text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma :\\ &\supset . \text{D}ʻʻ\lambda \text{ sm } \text{ sm } (\gamma \downarrow_{,,}ʻʻ\lambda ) . (\text{D}ʻ\dot{s} ʻ\lambda ) \text{ sm } (\lambda \times \gamma ) . ({\in}_{\Delta}ʻ\text{D}ʻʻ\lambda ) \text{ sm } (\gamma \,\,\text{exp}\,\, \lambda ) .\\ &\text{Nc}ʻ\text{Prod}ʻ\text{D}ʻʻ\lambda = \Pi \text{Nc}ʻ\text{D}ʻʻ\lambda = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\lambda }\\ &[*116·414·25 . *115·51 . *111·44 . *41·43]\end{align}\]

*116·422. \[\begin{align}&\vdash \colon\ldotp M \upharpoonright \delta \in 1 \rightarrow 1 : w, v \in \delta . \exists ! \text{D}ʻMʻw \cap \text{D}ʻMʻv . \supset _{w,v} . w = v :\\ &w \in \delta . \supset _{w} . Mʻw \in 1 \rightarrow 1 . \text{ᗡ}ʻMʻw = \gamma : \supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻ\delta\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *116·42 \frac{Mʻʻ\delta}{\lambda}. \supset \\ \vdash \colon\ldotp Mʻʻ\delta \subset 1 \rightarrow 1 : R, S \in Mʻʻ\delta . \exists ! \text{D}ʻR \cap \text{D}ʻS . &\supset _{R, S} . R = S : \text{ᗡ}ʻʻMʻʻ\delta \subset \iota ʻ\gamma :\\ &\supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻMʻʻ\delta &\qquad \text{(1)}\\ \vdash . *14·21 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : w \in \delta . \supset . \text{E}! Mʻw : &\qquad \text{(2)}\\ [*33·43] &\supset : \delta \subset \text{ᗡ}ʻM :\\ [*73·15] &\supset : (Mʻʻ\delta ) \text{ sm } \delta &\qquad \text{(3)}\\ \vdash . *51·15 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : w \in \delta . \supset . \text{ᗡ}ʻMʻw \in \iota ʻ\gamma :\\ [*37·61] &\supset : \text{ᗡ}ʻʻMʻʻ\delta \subset \iota ʻ\gamma &\qquad \text{(4)}\\ \vdash . (2) . *30·37 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : w,v \in \delta . \exists ! \text{D}ʻMʻw \cap \text{D}ʻMʻv . \supset _{w,v} . Mʻw = Mʻv :\\ [*37·63] &\supset : R, S \in Mʻʻ\delta . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R, S} . R = S &\qquad \text{(5)}\\ \vdash . (1) . (4) . (5) . \supset \vdash : \text{Hp} . &\supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻMʻʻ\delta .\\ [(3) . *113·13] &\supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻ\delta : \supset \vdash . \text{Prop} \end{array} \]

[Pg 156]

*116·43. \[\begin{align}&\vdash \colon\ldotp M\upharpoonright \delta \in 1\rightarrow 1:w,v\in \delta .\exists !\text{D}ʻMʻw\cap \text{D}ʻMʻv.\supset _{w,v}.w = v:\\ &w\in \delta .\supset _{w}.Mʻw\in 1\rightarrow 1.\text{ᗡ}ʻMʻw = \gamma :\\ &\supset .\text{Prod}ʻ\text{D}ʻʻMʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta ).\Pi \text{Nc}ʻ\text{D}ʻʻMʻʻ\delta = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta }\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*115·51.*116·422.&\supset \vdash :\text{Hp}.\supset .\text{Prod}ʻ\text{D}ʻʻMʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta ) &\qquad \text{(1)}\\ \vdash .*116·422.*114·52.&\supset \vdash :\text{Hp}.\supset .\Pi \text{Nc}ʻ\text{D}ʻʻMʻʻ\delta = \Pi \text{Nc}ʻ\gamma \downarrow_{,,}ʻʻ\delta &\qquad \text{(2)}\\ \vdash .*116·14·25.&\supset \vdash .\Pi \text{Nc}ʻ\gamma \downarrow_{,,}ʻʻ\delta = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

The above proposition is used in *116·534·61.

*116·44. \[\begin{align}&\vdash \colon\ldotp \exists !\gamma :(z).Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \text{V}:\\ &w,v\in \delta .\exists !(Mʻw)ʻʻ\gamma \cap (Mʻv)ʻʻ\gamma .\supset _{w,v}.w = v:\\ &\supset .\text{D}ʻʻ\upharpoonright \gammaʻʻMʻʻ\delta \text{ sm }\text{ sm }\gamma \downarrow_{,,}ʻʻ\delta .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta )\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*71·29.*35·65.\supset\\ \vdash \colon\ldotp \text{Hp}:(z).Nʻz = (Mʻz)\upharpoonright \gamma :\supset .(z).Nʻz\in 1\rightarrow 1.\text{ᗡ}ʻNʻz = \gamma &\qquad \text{(1)}\\ \vdash .*37·401.\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).\supset :w,v\in \delta .\exists !\text{D}ʻNʻw\cap \text{D}ʻNʻv.\supset _{w,v}.w = v &\qquad \text{(2)}\\ \vdash .*35·7.\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).\supset :x\in \gamma .w,v\in \delta .Nʻw = Nʻv.\supset .(Nʻw)ʻx = (Nʻv)ʻx.\\ [(2)] \supset .w = v &\qquad \text{(3)}\\ \vdash .(3).*10·11·23·35.\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).\supset :w,v\in \delta .Nʻw = Nʻv.\supset .w = v:\\ [*71·55·166] \supset :N\upharpoonright \delta \in 1\rightarrow 1 &\qquad \text{(4)}\\ \vdash .(1).(2).(4).*116·422.*115·51.\supset\\ \vdash :\text{Hp}.\text{Hp}(1).\supset .\text{D}ʻʻNʻʻ\delta \text{ sm }\text{ sm }\gamma \downarrow_{,,}ʻʻ\delta .\text{Prod}ʻ\text{D}ʻʻNʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta ) &\qquad \text{(5)}\\ \vdash .*38·11.\supset \vdash :\text{Hp}.\text{Hp}(1).\supset .\text{D}ʻNʻz = \text{D}ʻ\upharpoonright \gamma ʻMʻz.\\ [*37·353] \supset .\text{D}ʻʻNʻʻ\delta = \text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta &\qquad \text{(6)}\\ \vdash .(5).(6).\supset \vdash .\text{Prop} \end{array} \]

*116·45. \[\begin{align}&\vdash \colon\ldotp (z).Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \text{V}:\\ &w,v\in \delta .\exists !(Mʻw)ʻʻ\gamma \cap (Mʻv)ʻʻ\gamma .\supset _{w,v}.w\\ & = v:\supset .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta \text{ sm }(\gamma \,\,\text{exp}\,\,\delta )\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*116·182.*115·142.*37·29.\supset\\ \vdash :\text{Hp}.\gamma &= \Lambda .\exists !\delta .\supset .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta = \Lambda .\gamma \,\,\text{exp}\,\,\delta = \Lambda &\qquad \text{(1)}\\ \vdash .*115·1.*83·15.*116·181.\supset\\ \vdash :\text{Hp}.\delta & = \Lambda .\supset .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta = \iota ʻ\Lambda .\gamma \,\,\text{exp}\,\,\delta = \iota ʻ\Lambda &\qquad \text{(2)}\\ \vdash .(1).(2).*116·44.\supset \vdash .\text{Prop} \end{array} \]

The above proposition is used in *116·676.

We have now to prove the three formal laws of exponentiation, namely \[ \begin{aligned} \mu ^\nu \times _{c} \mu ^{\varpi } &= \mu ^{\nu +_{c} \varpi },\\ \mu ^{\varpi } \times _{c} \nu ^{\varpi } &= (\mu \times _{c} \nu )^{\varpi },\\ \text{and}\qquad (\mu ^\nu )^{\varpi } &= \mu ^{\nu \times _{c} \varpi }. \end{aligned} \][Pg 157] Of these the first is an immediate consequence of the distributive law, while the second and third result from forms of the associative law of multiplication.

*116·5. \(\vdash : \beta \cap \gamma = \Lambda . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma )\)

Dem. \[ \begin{array}{l} \vdash . *113·191 . \supset \\ \vdash : \text{Hp} . \exists ! \alpha . &\supset . \alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma = \Lambda .\\ [*114·301] &\supset . {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta \times {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻ(\alpha \downarrow_{,,}ʻʻ\beta \cup \alpha \downarrow_{,,}ʻʻ\gamma .\\ [*116·14 . *113·13] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } {\in}_{\Delta}ʻ(\alpha \downarrow_{,,}ʻʻ\beta \cup \alpha \downarrow_{,,}ʻʻ\gamma ) .\\ [*37·22] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ(\beta \cup \gamma ) .\\ [*116·14] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) &\qquad \text{(1)}\\ \vdash . *116·182 . &\supset \vdash : \alpha = \Lambda . \exists ! \beta . \supset . \alpha \,\,\text{exp}\,\, \beta = \Lambda .\\ [*113·114] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) = \Lambda &\qquad \text{(2)}\\ \vdash . *116·182 . *24·56 . &\supset \vdash : \alpha = \Lambda . \exists ! \beta . \supset . \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) = \Lambda &\qquad \text{(3)}\\ \vdash . (2) . (3) . &\supset \vdash : \alpha = \Lambda . \exists ! \beta . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) &\qquad \text{(4)}\\ \text{Similarly}\quad &\vdash : \alpha = \Lambda . \exists ! \gamma . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\,(\beta \cup \gamma ) &\qquad \text{(5)}\\ \vdash . *116·181 . &\supset \vdash : \alpha = \Lambda . \beta = \Lambda . \gamma = \Lambda . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) = \iota ʻ\Lambda \times \iota ʻ\Lambda &\qquad \text{(6)}\\ \vdash . *116·181 . &\supset \vdash : \alpha = \Lambda . \beta = \Lambda . \gamma = \Lambda . \supset . \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) = \iota ʻ\Lambda &\qquad \text{(7)}\\ \vdash . (6) . (7) . *113·611 . *73·43 . \supset \\ &\vdash : \alpha = \Lambda . \beta = \Lambda . \gamma = \Lambda . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) &\qquad \text{(8)}\\ \vdash . (1) . (4) . (5) . (8) . \supset \vdash . \text{Prop} \end{array} \]

In the last line of the above proof, *73·43 is required because the two \(\Lambda\)'s involved have not been proved to be of the same type. They are in fact of the same type, but it is unnecessary to prove this.

*116·51. \(\vdash . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta + \gamma )\)

Dem. \[ \begin{array}{l} \vdash . *116·19 . *110·12 . \supset \vdash . &(\alpha \,\,\text{exp}\,\, \beta ) \text{ sm } (\alpha \,\,\text{exp}\,\, \downarrow \Lambda _{\gamma } ʻʻ\iota ʻʻ\beta ) .\\ &(\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } (\alpha \,\,\text{exp}\,\, \Lambda _{\beta } \downarrow ʻʻ\iotaʻʻ\gamma ) .\\ [*113·13] \supset \vdash . &(\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm }\\ &(\alpha \,\,\text{exp}\,\, \downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta ) \times (\alpha \,\,\text{exp}\,\, \Lambda _{\beta } \downarrow ʻʻ\iotaʻʻ\gamma ) .\\ [*110·11 . *116·5] \supset \vdash . &(\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm }\\ &\alpha \,\,\text{exp}\,\, (\downarrow \Lambda _{\gamma } ʻʻ\iotaʻʻ\beta \cup \Lambda _{\beta } \downarrow ʻʻ\iotaʻʻ\gamma ) &\qquad \text{(1)}\\ \vdash . (1) . (*110·01) . \supset \vdash . \text{Prop} \end{array} \]

[Pg 158]

*116·52. \(\vdash .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{v +_{c}\varpi }\)

Dem. \[ \begin{array}{l} \vdash .*116·51 .*110·22.\supset \\ \vdash .(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta } \times _{c} (\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\gamma } = (\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta +_{c} \text{N}_{0}\text{c}ʻ\gamma } &\qquad \text{(1)}\\ \vdash .(1).*103·2.\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{\nu +_{c}\varpi } &\qquad \text{(2)}\\ \vdash .*116·205.*113·204.\supset \\ \vdash : \mu {\sim} \in \text{N}_{0}\text{C}.\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \Lambda = \mu ^{\nu +_{c}\varpi } &\qquad \text{(3)}\\ \vdash .*116·205.*113·204.\supset\\ \vdash : {\sim} (\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \Lambda &\qquad \text{(4)}\\ \vdash .*110·4.*116·204.\supset \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\mu ^{\nu +_{c}\varpi } = \Lambda &\qquad \text{(5)}\\ \vdash . (4).(5).\supset \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{\nu +_{c}\varpi } &\qquad \text{(6)}\\ \vdash .(2).(3).(6).\supset \vdash .\text{Prop} \end{array} \]

The following propositions are lemmas for \[ \mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi }. \] The principal previous propositions used in the proof are *115·6 and *116·43. The proof proceeds as follows. \[ (\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\,\text{is}\,\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\gamma \times \text{Prod}ʻ\beta \downarrow_{,,}ʻʻ\gamma. \] This, using *115·6, and putting \(\alpha \downarrow_{,,}\), \(\beta \downarrow_{,,}\) in place of \(R\) and \(S\) of that proposition, is similar to \[ {\in}_{\Delta}ʻ\hat{\mu} \{\exists z).z\in \gamma .\mu = \alpha \downarrow_{,,}z \times \beta \downarrow_{,,}z\},\, \textit{i.e.}\, \text{to}\, {\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\}. \] Now by *113·65, putting \(R\dagger = R\parallel \breve{R}\quad\text{Dft}\), \(\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta =(\downarrow z)\dagger ʻʻ(\alpha \times \beta)\). We now apply *116·43, taking \((\downarrow z)\dagger\) as the \(Mʻz\) of that proposition, or rather, taking \((\downarrow z)\dagger\upharpoonright (\alpha \times \beta)\). Thus we find \[ {\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \}\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma . \] Hence our proposition follows.

*116·529. \(R\dagger = R\parallel \breve{R} \quad\text{Dft}[*116]\)

In *150, this notation will be introduced as a permanent definition. For the present, we only introduce it to avoid \((\downarrow z\parallel\text{Cnv}ʻ\downarrow z)\), which is awkward.

*116·53. \[\begin{align}\vdash :\exists !&\alpha .\exists !\beta .\supset .\\ &(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*113·104·111.&\supset \vdash .\gamma \subset \text{ᗡ}ʻ\alpha \downarrow_{,,}.\gamma \subset \text{ᗡ}ʻ\beta \downarrow_{,,}.\alpha \downarrow_{,,}ʻʻ\gamma ,\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ \vdash .*113·105. &\supset \vdash :\text{Hp}.\supset .(\alpha \downarrow_{,,}ʻʻ\gamma )\upharpoonleft \alpha \downarrow_{,,}, \beta \downarrow_{,,}\upharpoonright \gamma \in 1\rightarrow 1 &\qquad \text{(2)}\\ \vdash .(1).(2).*115·6 \frac{\alpha \downarrow_{,,},\,\beta \downarrow_{,,}}{R,\,S}.\supset \\ \vdash : \text{Hp}.&\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }{\in}_{\Delta}ʻ\hat{\mu}\{(\exists z).z\in \gamma .\mu = \alpha \downarrow_{,,}z \times \beta \downarrow_{,,}z\} &\qquad \text{(3)}\\ \vdash .(3).*38·2.\supset \vdash .\text{Prop} \end{array} \]

[Pg 159]

The hypothesis \(\exists !\alpha .\exists !\beta\) is not necessary in the above proposition; but the proof is simpler with the hypothesis, and we do not need the proposition without the hypothesis.

*116·531. \[\begin{align}\vdash \colon\ldotp &M = \hat{R} \hat{z} \{z\in \gamma .R = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta )\}.\supset :\\ &z\in \gamma .\supset _{z}.Mʻz = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta ).Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \times \beta \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*74·772.*55·12.*72·184 .&\supset \vdash .(\downarrow z)\dagger \in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.z\in \gamma .\supset :RMz. \equiv .R = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta ):\\ [*30·3] &\supset :Mʻz = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta ): &\qquad \text{(2)}\\ [(1).*43·122] &\supset :Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \times \beta &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*116·532. \(\vdash :\text{Hp}*116·531.\exists !\alpha .\exists !\beta .\supset .M\in 1\rightarrow 1.\text{ᗡ}ʻM = \gamma\)

Dem. \[ \begin{array}{l} \vdash .*116·531.*14·21.*71·16.&\supset \vdash :\text{Hp}.\supset .M\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash . *116·531.\supset \vdash \colon\ldotp \text{Hp}.z,w\in \gamma .Mʻz = Mʻw.\supset :\\ &(\downarrow z)\dagger \upharpoonright (\alpha \times \beta ) = (\downarrow w)\dagger \upharpoonright (\alpha \times \beta ):\\ [*71·35] &\supset :R\in (\alpha \times \beta ).\supset .(\downarrow z)\dagger ʻR = (\downarrow w)\dagger ʻR:\\ [*113·101] \supset :x\in \alpha .y\in \beta .&\supset .(\downarrow z)\dagger ʻ(y\downarrow x) = (\downarrow w)\dagger ʻ(y\downarrow x).\\ [*113·123] &\supset .(y\downarrow z)\downarrow (x\downarrow z) = (y\downarrow w)\downarrow (x\downarrow w).\\ [*55·202] &\supset .z = w &\qquad \text{(2)}\\ \vdash .(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in \gamma .Mʻz = Mʻw.\supset .z = w &\qquad \text{(3)}\\ \vdash .*116·531.*14·21.*33·43.&\supset \vdash :\text{Hp}.z\in \gamma .\supset .z\in \text{ᗡ}ʻR &\qquad \text{(4)}\\ \vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :RMz.\supset _{R,z}.z\in \gamma :\\ [*33·351] &\supset :\text{ᗡ}ʻR\subset \gamma &\qquad \text{(5)}\\ \vdash .(1).(3).(4).(5).*71·55.\supset \vdash .\text{Prop} \end{array} \]

*116·533. \[\begin{align}\vdash \colon\ldotp \text{Hp}*116·531.\supset :\text{D}ʻʻMʻʻ\gamma = \hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\}:\\ z,w\in \gamma .\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻw.\supset _{z,w}.z = w\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*116·531.\supset \vdash :\text{Hp}.z\in \gamma .\supset .\text{D}ʻMʻz &= \text{D}ʻ\{(\downarrow z)\dagger \upharpoonright (\alpha \times \beta )\}\\ [*37·401] &= (\downarrow z)\dagger ʻʻ(\alpha \times \beta )\\ [*113·65] &= \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta &\qquad \text{(1)}\\ \vdash .(1).*37·6.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻʻMʻʻ\gamma = \hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\} &\qquad \text{(2)}\\ \vdash .*113·19.\supset \vdash :&\exists !(\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta )\cap (\downarrow wʻʻ\alpha \times \downarrow wʻʻ\beta ).\supset .\\ &\exists !\downarrow zʻʻ\alpha \cap \downarrow wʻʻ\alpha .\\ [*55·232] &\supset .z = w &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

[Pg 160]

*116·534. \(\vdash :\text{Hp}*116·532.\supset .{\in}_{\Delta}ʻ\text{D}ʻʻMʻʻ\gamma \text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma\)

Dem. \[ \begin{array}{l} \vdash .*116·531·532·533.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :M\in 1\rightarrow 1:z,w\in \gamma .&\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻw.\supset _{z,\omega }.z = w:\\ &z\in \gamma .\supset _{z}.Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \times \beta :\\ [*116·43] &\supset :\text{Prod}ʻ\text{D}ʻʻMʻʻ\gamma \text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma :\\ [*115·12.*30·37.*84·11]&\supset :{\in}_{\Delta}ʻ\text{D}ʻʻMʻʻ\gamma \text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\, \gamma \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*116·535. \[\begin{align}&\vdash :\exists !\alpha .\exists !\beta .\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma\\ &[*116·53·533·534]\end{align}\]

The hypothesis \(\exists !\alpha .\exists !\beta\) is not necessary, as we shall now prove.

*116·54. \(\vdash .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma \)

Dem. \[ \begin{array}{l} \vdash .*116·182.\supset \vdash :\alpha = \Lambda .\exists !\gamma .&\supset .\alpha \,\,\text{exp}\,\,\gamma = \Lambda .\\ [*113·114] &\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma ) = \Lambda &\qquad \text{(1)}\\ \vdash .*113·114.*116·182.&\supset \vdash :\alpha = \Lambda .\exists !\gamma .\supset .(\alpha \times \beta )\,\,\text{exp}\,\,\gamma = \Lambda &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash :\alpha = \Lambda .\exists !\gamma .\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(3)}\\ \text{Similarly}\quad &\vdash :\beta = \Lambda .\exists !\gamma .\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(4)}\\ \vdash .*116·181. \supset \vdash :\gamma = \Lambda .&\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma ) = \iota ʻ\Lambda \times \iota ʻ\Lambda .\\ [*113·611.*73·43] &\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }\iota ʻ\Lambda &\qquad \text{(5)}\\ \vdash .*116·181. \supset \vdash :\gamma = \Lambda .&\supset .(\alpha \times \beta )\,\,\text{exp}\,\,\gamma = \iota ʻ\Lambda .\\ [(5)] &\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(6)}\\ \vdash .(3).(4).(6).*116·535.&\supset \vdash .\text{Prop} \end{array} \]

In obtaining (5), we use *73·43 as well as *113·611, because \(\Lambda\)'s of different types are involved.

*116·55. \(\vdash .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi }\)

Dem. \[ \begin{array}{l} \vdash .*116·54·222.*113·222.&\supset \vdash .(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\gamma } \times _{c} (\text{N}_{0}\text{c}ʻ\beta )^{\text{N}_{0}\text{c}ʻ\gamma }\\ &=(\text{N}_{0}\text{c}ʻ\alpha \times _{c} \text{N}_{0}\text{c}ʻ\beta )^{\text{N}_{0}\text{c}ʻ\gamma } &\qquad \text{(1)}\\ \vdash .(1).*103·2. &\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi } &\qquad \text{(2)}\\ \vdash .*116·205.*113·204.&\supset \vdash :\varpi {\sim}\in \text{N}_{0}\text{C}.\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = \Lambda = (\mu \times _{c} \nu )^{\varpi } &\qquad \text{(3)}\\ \vdash .*116·205.*113·204.&\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = \Lambda &\qquad \text{(4)}\\ \vdash .*113·204.*116·204.&\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .(\mu \times _{c} \nu )^{\varpi } = \Lambda &\qquad \text{(5)}\\ \vdash .(4).(5) &\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi } &\qquad \text{(6)}\\ \vdash .(2).(3).(6). \supset \vdash .\text{Prop} \end{array} \]

This completes the proof of the second of the formal laws of exponentiation. The following propositions are lemmas for the third of these laws, namely \[ (\mu ^{\nu })^{\varpi } = \mu ^{\nu \times _{c} \varpi }. \]

[Pg 161]

*116·6. \[\begin{align}\vdash :\exists !\alpha .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm } \text{Prod}ʻ\text{Prod}ʻʻ&\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma .\\ &\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{3}\,\text{arithm}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*113·105.*84·53 \frac{\alpha \downarrow_{,,}}{R}.*113·111.&\supset \vdash :\text{Hp}.\supset .\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ \vdash .*40·38. &\supset \vdash .sʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma = \alpha \downarrow_{,,}ʻʻsʻ\beta \downarrow_{,,}ʻʻ\gamma &\qquad \text{(2)}\\ [*113·111] &\supset \vdash .sʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} &\qquad \text{(3)}\\ \vdash .(1).(3).*115·2. &\supset \vdash :\text{Hp}.\supset .\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{3}\,\text{arithm} &\qquad \text{(4)}\\ \vdash .*113·141.*116·19. \supset \vdash .&\text{Nc}ʻ\{\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\} = \text{Nc}ʻ\{\alpha \,\,\text{exp}\,\,(\gamma \times \beta )\}\\ [(*116·01.*113·02)] &= \text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻsʻ\beta \downarrow_{,,}ʻʻ\gamma \\ [(2)] &= \text{Nc}ʻ\text{Prod}ʻsʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \\ [*115·35.(4)] &= \text{Nc}ʻ\text{Prod}ʻ\text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma &\qquad \text{(5)}\\ \vdash .(4).(5).\supset \vdash .\text{Prop} \end{array} \]

*116·601. \(\vdash .\mid (\text{Cnv}ʻ\downarrow z)\in 1\rightarrow 1 \quad[*74·774.*72·184]\)

*116·602. \[\begin{align}\vdash \colon\ldotp M = \hat{R} \hat{z} &[z\in \gamma .R = \{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta )].\supset :\\ &z\in \gamma .\supset .Mʻz = \{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta ):\text{ᗡ}ʻM = \gamma \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*21·33.\supset \\ \vdash \colon\colon \text{Hp}.\supset \colon\ldotp z\in \gamma .\supset :RMz. \equiv .R = {\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta ) &\qquad \text{(1)}\\ \vdash .(1).*30·3.\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \gamma .\supset .Mʻz = {\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta ) &\qquad \text{(2)}\\ \vdash .*21·33.*33·131.\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻM = \gamma &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*116·603. \[\begin{align}&\vdash \colon\ldotp \text{Hp}*116·602.\supset :z\in \gamma .\supset .\text{ᗡ}ʻMʻz = \alpha \,\,\text{exp}\,\,\beta \\ &[*116·602.*37·231.*35·65]\end{align}\]

*116·604. \(\vdash \colon\ldotp \text{Hp}*116·602.\supset :z\in \gamma .\supset .\text{D}ʻMʻz = \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z\)

Dem. \[ \begin{array}{l} \vdash .*37·401.*116·602.\supset \\ \vdash :\text{Hp}.z\in \gamma .\supset .\text{D}ʻMʻz &= \{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in}ʻʻ(\alpha \,\,\text{exp}\,\,\beta )\\ [*115·4.*116·601.*43·301] &= \text{Prod}ʻ\{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in}ʻʻ\alpha \downarrow_{,,}ʻʻ\beta \\ [*113·125 \frac{I}{R}.*50·75·16] &= \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\downarrow zʻʻ\beta \\ [*38·2] &= \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z:\supset \vdash .\text{Prop} \end{array} \]

*116·605. \(\vdash \colon\ldotp \text{Hp}*116·602.\supset :z\in \gamma .\supset .Mʻz\in 1\rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash .*116·601.*72·451.\supset \\ \vdash .{\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻ\mid (\text{Cnv}ʻ\downarrow z)\in 1\rightarrow 1.\\ [*43·301]\supset \vdash .{\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta )\in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash .(1).*116·602.\supset \vdash .\text{Prop} \end{array} \]

[Pg 162]

*116·606. \[\begin{align}\vdash \colon\ldotp \text{Hp}*116·602.&\exists !\alpha .\exists !\beta .\supset :\\ &M\in 1\rightarrow 1:z,w\in \gamma .\text{D}ʻMʻz = \text{D}ʻMʻw.\supset _{z,w}.z = w\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*116·602.*14·21.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \text{ᗡ}ʻM.\supset _{z}.\text{E}!Mʻz:\\ [*71·16] &\supset :M\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash .*30·37.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in \gamma .Mʻz = Mʻw.\supset .\text{D}ʻMʻz = \text{D}ʻMʻw &\qquad \text{(2)}\\ \vdash .*116·604.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in \gamma .\text{D}ʻMʻz = \text{D}ʻMʻw.\supset .\\ &\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z = \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}w.\\ [*30·37] &\supset .sʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z = sʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}w.\\ [*116·171.*115·141.(*116·01)] &\supset .sʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z = sʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}w.\\ [*113·1] &\supset .\beta \downarrow_{,,}z \times \alpha = \beta \downarrow_{,,}w \times \alpha .\\ [*113·182] &\supset .\beta \downarrow_{,,}z = \beta \downarrow_{,,}w.\\ [*113·105.\text{Hp}] &\supset .z = w &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*71·55.*116·602.\supset \vdash .\text{Prop} \end{array} \]

*116·607. \[\begin{align}&\vdash \colon\ldotp \text{Hp}*116·602.\exists !\alpha .\exists !\beta .\supset :\\ &M\in 1\rightarrow 1.\text{D}ʻʻMʻʻ\gamma = \text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma :\\ &z,w\in \gamma .\text{D}ʻMʻz = \text{D}ʻMʻw.\supset _{z,w}.z = w:\\ &z\in \gamma .\supset _{z}.Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \,\,\text{exp}\,\,\beta \quad[*116·606·604·605·603]\end{align}\]

*116·61. \[\begin{align}&\vdash :\exists !\alpha .\exists !\beta .\supset .\text{Prod}ʻ\text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma \\ &[*116·607·43]\end{align}\]

*116·611. \(\vdash :\exists !\alpha .\exists !\beta .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma \quad[116·6·61]\)

*116·62. \(\vdash .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma\)

Dem. \[ \begin{array}{l} \vdash .*116·181.*113·114.&\supset \vdash :\beta = \Lambda .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma ) = \iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash .*116·181.&\supset \vdash :\beta = \Lambda .\supset .(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma = (\iota ʻ\Lambda )\,\,\text{exp}\,\,\gamma &\qquad \text{(2)}\\ \vdash .*116·33·25.&\supset \vdash .\text{Nc}ʻ\{(\iota ʻ\Lambda )\,\,\text{exp}\,\,\gamma\}= 1 &\qquad \text{(3)}\\ \vdash . (1).(2).(3).*55·22.*100·31.&\supset \vdash :\beta = \Lambda .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(4)}\\ \vdash . *113·107.*116·182.&\supset \vdash :\alpha = \Lambda .\exists !\beta .\exists !\gamma .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma ) = \Lambda &\qquad \text{(5)}\\ \vdash .*116·182.&\supset \vdash :\alpha = \Lambda .\exists !\beta .\exists !\gamma .\supset .(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma = \Lambda &\qquad \text{(6)}\\ \vdash .(5).(6).&\supset \vdash :\alpha = \Lambda .\exists !\beta .\exists !\gamma .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(7)}\\ \vdash .*113·114.*116·181.&\supset \vdash :\gamma = \Lambda .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma ) = \iota ʻ\Lambda .(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma = \iota ʻ\Lambda .\\ [*73·43] &\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(8)}\\ \vdash .(4).(7).(8).&\supset \vdash \colon\ldotp \alpha = \Lambda .\lor.\beta = \Lambda .\lor.\gamma = \Lambda :\supset .\\ &\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(9)}\\ \vdash .(9).*116·611.\supset \vdash .\text{Prop} \end{array} \]

[Pg 163]

*116·63. \(\vdash . \mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi }\)

Dem. \[ \begin{array}{l} \vdash . *113·222 . \supset \vdash . (\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta \times _{c} \text{N}_{0}\text{c}ʻ\gamma } &= (\text{N}_{0}\text{c}ʻ\alpha )^{\text{Nc}ʻ(\beta \times \gamma )}\\ [*116·222 . (*116·04)] &= \text{Nc}ʻ\{\alpha \,\,\text{exp}\,\, (\beta \times \gamma )\}\\ [*116·62] &= \text{Nc}ʻ\{(\alpha \,\,\text{exp}\,\, \beta ) \,\,\text{exp}\,\, \gamma\}\\ [*116·222] &= \{\text{N}_{0}\text{c}ʻ(\alpha \,\,\text{exp}\,\, \beta )\}^{\text{N}_{0}\text{c}ʻ\gamma }\\ [*116·222 . (*116· 03)] &= \{(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta }\}^{\text{N}_{0}\text{c}ʻ\gamma } &\qquad \text{(1)}\\ \vdash . (1) . *103·2 . &\supset \vdash : \mu , \nu , \varpi \in \text{N}_{0}\text{C} . \supset . \mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi } &\qquad \text{(2)}\\ \vdash . *116·204·205 . &\supset \vdash : {\sim} (\mu , \nu \in \text{N}_{0}\text{C}) . \supset . (\mu ^\nu )^{\varpi } = \Lambda &\qquad \text{(3)}\\ \vdash . *113·205 . *116·204·205 . &\supset \vdash : {\sim}(\mu , \nu \in \text{N}_{0}\text{C}) . \supset . \mu ^{\nu \times _{c} \varpi } = \Lambda &\qquad \text{(4)}\\ \vdash . *116·205 . &\supset \vdash : \varpi {\sim} \in \text{N}_{0}\text{C} . \supset . (\mu ^\nu )^{\varpi } = \Lambda &\qquad \text{(5)}\\ \vdash . *113·205 . *116·204 . &\supset \vdash : \varpi {\sim} \in \text{N}_{0}\text{C} . \supset . \mu ^{\nu \times _{c} \varpi } = \Lambda &\qquad \text{(6)}\\ \vdash . (3) . (4) . (5) . (6) . &\supset \vdash : {\sim} (\mu , \nu , \varpi \in \text{N}_{0}\text{C}) . \supset . \mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi } &\qquad \text{(7)}\\ \vdash . (2) . (7) . \supset \vdash . \text{Prop} \end{array} \]

This completes the proof of the third of the formal laws of exponentiation.

*116·64. \(\vdash . (\mu ^\nu )^{\varpi } = (\mu ^{\varpi })^\nu \quad[*116·63 . *113·27]\)

*116·651. \(\vdash : Q \in \text{Cls} \rightarrow 1 . \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻP_{\Delta }ʻʻQʻʻʻ\kappa \text{ sm } P_{\Delta }ʻQʻʻsʻ\kappa \)

Dem. \[ \begin{array}{l} \vdash . *84·53 . \supset \vdash : \text{Hp} . &\supset . Qʻʻʻ\kappa \in \text{Cls}^{2} \text{excl}.\\ [*85·43] &\supset . {\in}_{\Delta}ʻP_{\Delta }ʻʻQʻʻʻ\kappa \text{ sm } P_{\Delta }ʻsʻQʻʻʻ\kappa .\\ [*40·38] &\supset . {\in}_{\Delta}ʻP_{\Delta }ʻʻQʻʻʻ\kappa \text{ sm } P_{\Delta }ʻQʻʻsʻ\kappa : \supset \vdash . \text{Prop} \end{array} \]

*116·652. \[\begin{align}&\vdash : Q \in \text{Cls} \rightarrow 1 . \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻQʻʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻQʻʻsʻ\kappa \\ &\left[*116·651 \frac{\in}{P}\right]\end{align}\]

The following propositions are lemmas for *116·661, which is an extension of *116·52.

*116·653. \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{arithm}\)

Dem. \[ \begin{array}{l} \vdash . *113·105 . *84·53 . &\supset \vdash : \text{Hp} . \exists ! \alpha . \supset . \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ \vdash . *113·111 . &\supset \vdash . \alpha \downarrow_{,,}ʻʻsʻ\kappa \in \text{Cls}^{2} \text{excl} .\\ [*40·38] &\supset \vdash . sʻ\alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} . &\qquad \text{(2)}\\ \vdash . *113·112·113 . \supset \vdash \colon\ldotp \alpha = \Lambda . \supset : &\beta \in \kappa . \exists ! \beta . \supset . \alpha \downarrow_{,,}ʻʻ\beta = \iota ʻ\Lambda :\\ &\beta \in \kappa . \beta = \Lambda . \supset . \alpha \downarrow_{,,}ʻʻ\beta = \Lambda &\qquad \text{(3)}\\ \vdash . (3) . \supset \vdash \colon\ldotp \alpha = \Lambda . &\supset : \alpha \downarrow_{,,}ʻʻʻ\kappa \subset \iota ʻ\iota ʻ\Lambda \cup \iota ʻ\Lambda :\\ [*24·43·561] &\supset : \rho , \sigma \in \alpha \downarrow_{,,}ʻʻʻ\kappa . \exists ! \rho \cap \sigma . \supset . \rho , \sigma \in \iota ʻ\iota ʻ\Lambda .\\ [*51·15] &\supset . \rho = \sigma :\\ [*84·11] &\supset : \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(4)}\\ \vdash . (1) . (4) . &\supset \vdash : \text{Hp} . \supset . \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(5)}\\ \vdash . (2) . (5) . \supset \vdash . \text{Prop} \end{array} \]

[Pg 164]

*116·654. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\{\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \}\text{ sm }\{\alpha \,\,\text{exp}\,\,(sʻ\kappa )\}\)

Dem. \[ \begin{array}{l} \vdash .*38·13.(*116·01).\supset \vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = \text{Prod}ʻ\text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\kappa &\qquad \text{(1)}\\ \vdash .(1).*116·653.*115·34.\supset \\ \vdash .\{\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \}\text{ sm }\{\text{Prod}ʻsʻ\alpha \downarrow_{,,}ʻʻʻ\kappa\} &\qquad \text{(2)}\\ \vdash .(2).*40·38.(*116·01).\supset \vdash .\text{Prop} \end{array} \]

*116·655. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Pi \text{Nc}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = (\text{Nc}ʻ\alpha )^{\Sigma \text{Nc}ʻ\kappa } \quad[*116·654]\)

This proposition is an extension of *116·5.

The hypothesis \(\kappa \in \text{Cls}^{2} \text{excl}\) is unnecessary in the above proposition, as we shall now prove.

*116·656. \(\vdash :\exists !\alpha \,\,\text{exp}\,\,\beta \cap \alpha \,\,\text{exp}\,\,\gamma .\supset .\beta = \gamma \)

Dem. \[ \begin{array}{l} \vdash .*116·11.*52·16.\supset \\ \vdash \colon\ldotp \mu \in (\alpha \,\,\text{exp}\,\, \beta )\cap (\alpha \,\,\text{exp}\,\, \gamma ).&\supset :y\in \beta .\supset .(\exists x).x\in \alpha .x\downarrow y\in \mu :\mu \subset \gamma \times \alpha :\\ [*113·101] &\supset :y\in \beta .\supset .(\exists x).x\in \alpha .x\downarrow y\in \mu :x\downarrow y\in \mu .\supset .y\in \gamma :\\ [\text{Syll}] &\supset :y\in \beta .\supset .(\exists x).x\in \alpha .x\downarrow y\in \mu .y\in \gamma .\\ [*10·35] &\supset .y\in \gamma &\qquad \text{(1)}\\ \text{Similarly}\quad &\vdash \colon\ldotp \mu \in \alpha \,\,\text{exp}\,\, \beta \cap \alpha \,\,\text{exp}\,\, \gamma .\supset :y\in \gamma .\supset .y\in \beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*116·657. \(\vdash .(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \in \text{Cls}^{2} \text{excl} \quad[*116·656]\)

*116·658. \(\vdash .\alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\beta ) = {\mid (\text{Cnv}ʻ\downarrow \beta )}_{\in}ʻʻ(\alpha \,\,\text{exp}\,\, \beta )\)

Dem. \[ \begin{array}{l} \vdash .*116·602·604.*37·401.\supset \vdash .\{\mid (\text{Cnv}ʻ\downarrow \beta )\}_{\in}ʻʻ(\alpha \,\,\text{exp}\,\,\beta ) &= \alpha \,\,\text{exp}\,\,(\beta \downarrow_{,,}\beta )\\ [*85.601] &= \alpha \,\,\text{exp}\,\,(\in \unicode{x21A7}\beta ).\supset \vdash .\text{Prop} \end{array} \]

*116·659. \[\begin{align}T = \hat{\nu} \hat{\mu}\{(\exists \beta ).\beta \in \kappa .\mu \in \alpha \,\,\text{exp}\,\,\beta .&\nu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu\}.\supset .\\ &T\in (\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\alpha \,\,\text{exp}\,\,)ʻʻ\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*40·4. \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻT = sʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa &\qquad \text{(1)}\\ \vdash .*21·33.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu T\mu .\varpi T\mu .\supset .\\ &(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mu \in \alpha \,\,\text{exp}\,\,\beta .\mu \in \alpha \,\,\text{exp}\,\,\gamma .\\ &\nu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu .\varpi = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\mu .\\ [*116·656] &\supset .(\exists \beta ,\gamma ).\beta = \gamma .\nu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu .\varpi = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\mu .\\ [*13·195] &\supset .\nu = \varpi &\qquad \text{(2)}\\ \vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\varpi T\mu .\varpi T\nu .\supset .\\ &(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mu \in \alpha \,\,\text{exp}\,\,\beta .\nu \in \alpha \,\,\text{exp}\,\,\gamma .\\ &\varpi = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\nu .\\ [*116·658] &\supset .(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mu \in \alpha \,\,\text{exp}\,\, \beta .\nu \in \alpha \,\,\text{exp}\,\, \gamma .\\ &\varpi = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\nu .\\ &\varpi \in \alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\beta )\cap \alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\gamma ).\\ [*116·656] & \supset .(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\nu .\\ &\in \unicode{x21A7}\beta = \in \unicode{x21A7}\gamma .\\ [*85·601] &\supset .(\exists \beta ).\beta \in \kappa .\mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\nu .\\ [*116·601.*72·441] &\supset .\mu = \nu &\qquad \text{(3)}\\ \vdash .(2).(3). &\supset \vdash :\text{Hp}.\supset .T\in 1\rightarrow 1 &\qquad \text{(4)}\\ \vdash .*116·658.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \kappa .\supset .Tʻʻ(\alpha \,\,\text{exp}\,\, \beta ) = \alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\beta ):\\ [*37·69] &\supset :\tau _{\in }ʻʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = (\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa &\qquad \text{(5)}\\ \vdash .(1).(4).(5).*111·1 .\supset \vdash .\text{Prop} \end{array} \]

[Pg 165]

*116·66. \(\vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,)ʻʻ\kappa \text{ sm } \{\alpha \,\,\text{exp}\,\, (\Sigma ʻ\kappa )\}\)

Dem. \[ \begin{array}{l} \vdash .*116·659.*115·51. &\supset \vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \text{ sm } \text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa &\qquad \text{(1)}\\ \vdash .*85·61.*116·654. &\supset \vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa \text{ sm } \{\alpha \,\,\text{exp}\,\, (sʻ\in \unicode{x21A7}ʻʻ\kappa )\} &\qquad \text{(2)}\\ \vdash .(1).(2).*112·1. \supset \vdash .\text{Prop} \end{array} \]

*116·661. \(\vdash .\Pi \text{Nc}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = (\text{Nc}ʻ\alpha )^{\Sigma \text{Nc}ʻ\kappa } \quad[*116·66·657.*115·12.*112·101]\)

This proposition is an extension of *116·52.

The following propositions are concerned in proving *116·68, which is an extension of *116·54, where the \(\alpha\) and \(\beta\) of that proposition are replaced by the members of a class \(\kappa\).

*116·67. \(\vdash \colon\ldotp \rho = \hat{\lambda} \{(\exists \alpha ).\alpha \in \kappa .\lambda = \alpha \downarrow ʻʻ\gamma\}.\supset :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\rho \in \text{Cls}^{3}\,\text{arithm}\)

Dem. \[ \begin{array}{l} \vdash .*20·3 .\supset \vdash :&\text{Hp}.\lambda ,\mu \in \rho .\exists !\lambda \cap \mu .\supset .\\ &(\exists \alpha ,\beta ).\alpha ,\beta \in \kappa .\lambda = \alpha \downarrow_{,,}ʻʻ\gamma .\mu = \beta \downarrow_{,,}ʻʻ\gamma .\exists !\lambda \cap \mu .\\ [*37·6] &\supset .(\exists \alpha ,\beta ,z,w).\alpha ,\beta \in \kappa .z,w\in \gamma .\alpha \downarrow_{,,}z = \beta \downarrow_{,,}w.\lambda = \alpha \downarrow_{,,}ʻʻ\gamma .\\ &\mu = \beta \downarrow_{,,}ʻʻ\gamma .\\ [*55·262.*38·2]&\supset .(\exists \alpha ,z,w).\alpha \in \kappa .z,w\in \gamma .\lambda = \alpha \downarrow_{,,}ʻʻ\gamma .\mu = \alpha \downarrow_{,,}ʻʻ\gamma .\\ [*13·172] &\supset .\lambda =\mu &\qquad \text{(1)}\\ \vdash .*37·6.*40·11.\supset \\ \vdash :\text{Hp}.\xi ,\eta \in sʻ\rho .\exists !\xi \cap \eta .\supset .\\ &(\exists \alpha ,\beta ,z,w).\alpha ,\beta \in \kappa .z,w\in \gamma .\xi = \alpha \downarrow_{,,}z.\eta = \beta \downarrow_{,,}w.\exists !\xi \cap \eta .\\ [*55·232.*38·2]&\supset .(\exists \alpha ,\beta ,z).\alpha ,\beta \in \kappa .z\in \gamma .\xi = \alpha \downarrow_{,,}z.\eta = \beta \downarrow_{,,}z.\exists !\alpha \cap \beta &\qquad \text{(2)}\\ \vdash .(2).*84·11.\supset \\ \vdash :\text{Hp}.\kappa \in \text{Cls}^{2} \text{excl}.\xi ,\eta \in sʻ\rho .\exists !\xi \cap \eta .&\supset .(\exists \alpha ,\beta ,z).\xi = \alpha \downarrow_{,,}z.\eta = \beta \downarrow_{,,}z.\alpha = \beta .\\ [*13·195·172] &\supset :\xi = \eta &\qquad \text{(3)}\\ \vdash .(1).(3).\supset \vdash .\text{Prop} \end{array} \]

[Pg 166]

*116·671. \(\vdash \colon\ldotp \sigma = \hat{\mu} \{(\exists z) . z \in \gamma . \mu = \downarrow zʻʻʻ\kappa\} . \supset : \text{Hp} *116·67 . \supset . sʻ\rho = sʻ\sigma\)

Dem. \[ \begin{array}{l} \vdash . *40·11 . \supset \vdash \colon\ldotp \text{Hp} *116·67 . \supset : \xi \in sʻ\rho . &\equiv . (\exists \alpha ) . \alpha \in \kappa . \xi \in \alpha \downarrow_{,,} ʻʻ\gamma .\\ [*38·3] &\equiv . (\exists \alpha , z) . \alpha \in \kappa . z \in \gamma . \xi = \downarrow zʻʻ\alpha .\\ [*37·103] &\equiv . (\exists z) . z \in \gamma . \xi \in \downarrow zʻʻʻ\kappa .\\ [*40·11] &\equiv . \xi \in sʻ\hat{\mu}\{(\exists z) . z \in \gamma . \mu = \downarrow zʻʻʻ\kappa\} \colon\ldotp \\ \supset \vdash . \text{Prop} \end{array} \]

*116·672. \(\vdash : \text{Hp} *116·671 . \kappa \in \text{Cls}^{2} \text{excl} . \Lambda {\sim} \in \kappa . \supset . \sigma \in \text{Cls}^{2} \text{excl}\)

Dem. \[ \begin{array}{l} \vdash . *37·103 . \supset \vdash : &\text{Hp} . \mu , \nu \in \sigma . \exists ! \mu \cap \nu . \supset .\\ &(\exists z, w, \alpha , \beta ) . z, w \in \gamma . \alpha , \beta \in \kappa . \downarrow zʻʻ\alpha = \downarrow wʻʻ\beta .\\ \mu = \downarrow zʻʻʻ\alpha . \nu = \downarrow wʻʻʻ\beta .\\ [*55·262] &\supset . (\exists z, w, \alpha ) . z, w \in \gamma . \alpha \in \kappa . \downarrow zʻʻ\alpha = \downarrow wʻʻ\alpha .\\ \mu = \downarrow zʻʻʻ\alpha . \nu = \downarrow wʻʻʻ\alpha .\\ [*113·105 . *38·2 . \text{Hp}] &\supset . (\exists z, \alpha ) . \mu = \downarrow zʻʻʻ\alpha . \nu = \downarrow zʻʻʻ\alpha .\\ [*13·172] &\supset . \mu = \nu : \supset \vdash . \text{Prop} \end{array} \]

*116·673. \(\vdash : \text{Hp} *116·672 . \supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\sigma\)

Dem. \[ \begin{array}{l} \vdash . *38·131 . (*116·01) . \supset \vdash . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa &= {\in}_{\Delta}ʻ\hat{\xi} \{(\exists \alpha ) . \alpha \in \kappa . \xi = \text{Prod}ʻ\alpha \downarrow_{,,} ʻʻ\gamma\}\\ [*37·6] &= {\in}_{\Delta}ʻ\text{Prod}ʻʻ\hat{\lambda} \{(\exists \alpha ) . \alpha \in \kappa . \lambda = \alpha \downarrow_{,,} ʻʻ\gamma\} &\qquad \text{(1)}\\ \vdash . (1) . *115·33 . *116·67 . \supset \\ \vdash : \text{Hp} . &\supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻsʻ\hat{\lambda} \{(\exists \alpha ) . \alpha \in \kappa . \lambda = \alpha \downarrow_{,,} ʻʻ\gamma\} .\\ [*116·671] &\supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻsʻ\sigma .\\ [*85·44 . *116·672] &\supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\sigma : \supset \vdash . \text{Prop} \end{array} \]

*116·674. \[\begin{align}\vdash \colon\ldotp &M = \hat{R} \hat{z} \{R = (\downarrow z) \parallel \text{Cnv}ʻ(\downarrow z)_{\in }\} . \supset :\\ &(z) . Mʻz \in 1 \rightarrow 1 . \text{D}ʻ(Mʻz) \upharpoonright {\in}_{\Delta}ʻ\kappa = {\in}_{\Delta}ʻ \downarrow zʻʻʻ\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *30·3 . &\supset \vdash : \text{Hp} . \supset . Mʻz = (\downarrow z) \parallel \text{Cnv}ʻ(\downarrow z)_{\in } &\qquad \text{(1)}\\ \vdash . *72·184 . *111·14 . &\supset \vdash . \downarrow z \upharpoonright \kappa \in (\downarrow zʻʻʻ\kappa ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa .\\ [*114·51] &\supset \vdash . {\downarrow z \parallel \text{Cnv}ʻ(\downarrow z)_{\in }} \upharpoonright {\in}_{\Delta}ʻ\kappa \in ({\in}_{\Delta}ʻ \downarrow zʻʻʻ\kappa ) \,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\kappa ) &\qquad \text{(2)}\\ \vdash . (1) . (2) . *73·03 . &\supset \vdash . \text{Prop} \end{array} \]

*116·675 . \(\vdash \colon\ldotp \text{Hp} *116·674 . \exists ! sʻ\kappa . \supset : \exists ! (Mʻw)ʻʻ{\in}_{\Delta}ʻ\kappa \cap (Mʻ\nu )ʻʻ{\in}_{\Delta}ʻ\kappa . \supset . w = v\)

Dem. \[ \begin{array}{l} \vdash . *116·674 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : \exists ! (Mʻw)ʻʻ{\in}_{\Delta}ʻ\kappa \cap (Mʻv)ʻʻ{\in}_{\Delta}ʻ\kappa . \supset .\\ &\exists ! {\in}_{\Delta}ʻ \downarrow wʻʻʻ\kappa \cap {\in}_{\Delta}ʻ \downarrow vʻʻʻ\kappa .\\ [*80·32] &\supset . \downarrow wʻʻʻ\kappa = \downarrow vʻʻʻ\kappa .\\ [*40·38] &\supset . \downarrow wʻʻsʻ\kappa = \downarrow vʻʻsʻ\kappa .\\ [*113·105 . *38·2] &\supset . w = v \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

[Pg 167]

*116·676. \[\begin{align}\vdash : \text{Hp} *116·672·675 . \supset . \text{Prod}ʻ&\text{D}ʻʻ\upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma \text{ sm } ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma .\\ &\text{D}ʻʻ \upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma = {\in}_{\Delta}ʻʻ\sigma\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *116·674·675 . *116·45 \frac{{\in}_{\Delta}ʻ\kappa,\,\gamma} {\gamma,\,\delta}. \supset \\ \vdash : \text{Hp} . &\supset . \text{Prod}ʻ\text{D}ʻʻ \upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma \text{ sm } ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma &\qquad \text{(1)}\\ \vdash . *116·674 . &\supset \vdash : \text{Hp} . \supset . \text{D}ʻʻ \upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma = \hat{\mu} \{(\exists z) . z \in \gamma . \mu = {\in}_{\Delta}ʻ \downarrow zʻʻʻ\kappa\}\\ [*37·6 . \text{Hp}] &= {\in}_{\Delta}ʻʻ\sigma &\qquad \text{(2)}\\ \vdash . (1) . (2) . \supset \vdash . \text{Prop} \end{array} \]

*116·68. \[\begin{align}\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } &({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma .\\ &\Pi \text{Nc}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa = (\Pi \text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *115·12 . *84·55 . &\supset \vdash . \text{Prod}ʻ{\in}_{\Delta}ʻʻ\sigma \text{ sm } {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\sigma &\qquad \text{(1)}\\ \vdash . (1) . *116·673·676 . \supset \\ \vdash : \text{Hp} . \Lambda {\sim} \in \kappa . \exists ! sʻ\kappa . &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma)ʻʻk \text{ sm } ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma &\qquad \text{(2)}\\ \vdash . *53·24 . \supset \vdash : \Lambda {\sim} \in \kappa . {\sim} \exists ! sʻ\kappa . &\supset . \kappa = \Lambda .\\ [*83·15 . *116·33] &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa = \iota ʻ\dot{\Lambda} . ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma \in 1 .\\ [*73·45] &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma &\qquad \text{(3)}\\ \vdash . *83·11 . *116·182 . \supset \vdash : \Lambda \in \kappa . \exists ! \gamma . &\supset . {\in}_{\Delta}ʻ\kappa = \Lambda . \Lambda \in (\text{exp}\,\, \gamma )ʻʻ\kappa .\\ [*116·182 . *83·11] &\supset . ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma = \Lambda . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa = \Lambda &\qquad \text{(4)}\\ \vdash . *116·181 . &\supset \vdash : \Lambda \in \kappa . \gamma = \Lambda . \supset . ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma = \iota ʻ\Lambda &\qquad \text{(5)}\\ \vdash . *116·181 . &\supset \vdash : \Lambda \in \kappa . \gamma = \Lambda . \supset . (\text{exp}\,\, \gamma )ʻʻ\kappa = \iota ʻ\iota ʻ\Lambda .\\ [*83·41] &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } \iota ʻ\Lambda &\qquad \text{(6)}\\ \vdash . (4) . (5) . (6) . &\supset \vdash : \Lambda \in \kappa . \supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma &\qquad \text{(7)}\\ \vdash . (2) . (3) . (7) . *114·1 . *116·25 . \supset \vdash . \text{Prop} \end{array} \]

The above proposition is an extension of *116·54·55.

The following propositions are lemmas for \[ \text{Nc}ʻ\text{Cl}ʻ\alpha = 2^{\text{Nc}ʻ\alpha } . \]

The proposition and its proof are due to Cantor.

*116·7. \(\vdash . \text{Nc}ʻ\{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha \}_{\Delta }ʻ\alpha = 2^{\text{Nc}ʻ\alpha }\)

Dem. \[ \begin{array}{l} \vdash . *24·1 .*101·3 . \supset \vdash . \text{Nc}ʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V}) &= 2 &\qquad \text{(1)}\\ \vdash . *116·15 . \supset \vdash . \text{Nc}ʻ{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha &= \text{Nc}ʻ\{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \,\,\text{exp}\,\, \alpha\}\\ [*116·25] &= {\text{Nc}ʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V})}^{\text{Nc}ʻ\alpha }\\ [(1)] &= 2^{\text{Nc}ʻ\alpha } . \supset \vdash . \text{Prop} \end{array} \]

[Pg 168]

In this and following propositions, the class \(\iota ʻ\Lambda \cup\iota ʻ\text{V}\) is introduced solely as a known class consisting of two terms. Any other class of two terms will serve equally well.

*116·71. \(\vdash : R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha . \supset . \overleftarrow{R}ʻ\text{V} = \alpha - \overleftarrow{R}ʻ\Lambda \)

Dem. \[ \begin{array}{l} \vdash . *116·12 . \supset \vdash : \text{Hp} . &\supset . R \in 1 \rightarrow \text{Cls} . \text{D}ʻR \subset \iota ʻ\Lambda \cup \iota ʻ\text{V} . \text{ᗡ}ʻR = \alpha &\qquad \text{(1)}\\ [*37·271] \supset . \alpha &= \breve{R} ʻʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V})\\ [*53·302] &= \overleftarrow{R}ʻ\Lambda \cup \overleftarrow{R}ʻ\text{V} &\qquad \text{(2)}\\ \vdash . (1) . *71·18 . &\supset \vdash : \text{Hp} . \supset . \overleftarrow{R}ʻ\Lambda \cap \overleftarrow{R}ʻ\text{V} = \Lambda &\qquad \text{(3)}\\ \vdash . (2) . (3) . *24·47 . \supset \vdash . \text{Prop} \end{array} \]

*116·711. \(\vdash : R, S \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha . \overleftarrow{R}ʻ\Lambda = \overleftarrow{S}ʻ\Lambda . \supset . R = S\)

Dem. \[ \begin{array}{l} \vdash . *116·71 . &\supset \vdash : \text{Hp} . \supset . \overleftarrow{R}ʻ\text{V} = \overleftarrow{S}ʻ\text{V} &\qquad \text{(1)}\\ \vdash . (1) . *116·12 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \gamma \in \text{D}ʻR \cup \text{D}ʻS . \supset _{\gamma } . \overleftarrow{R}ʻ\gamma = \overleftarrow{S}ʻ\gamma :\\ [*33·48] &\supset : R = S \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

*116·712. \[\begin{align}&\vdash \colon\ldotp T = \hat{\mu} \hat{R} [R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha \}_{\Delta }ʻ\alpha . \mu = \overleftarrow{R}ʻ\Lambda ] . \supset :\\ &R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha . \supset . TʻR = \overleftarrow{R}ʻ\Lambda :\text{ᗡ}ʻT = \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *21·33 .& \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp R \in {(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha . \supset : \mu TR . \equiv _{\mu } . \mu = \overleftarrow{R}ʻ\Lambda :\\ [*30.3] &\supset : TʻR = \overleftarrow{R}ʻ\Lambda &\qquad \text{(1)}\\ \vdash . (1) . *14·21 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : R \in {(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha . \supset . \text{E}! TʻR .\\ [*33·43] &\supset . R \in \text{ᗡ}ʻT &\qquad \text{(2)}\\ \vdash . *21·33 . *33·131 . &\supset \vdash : \text{Hp} . \supset · \text{ᗡ}ʻT \subset {(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha &\qquad \text{(3)}\\ \vdash (1) . (2) . (3) . \supset \vdash . \text{Prop} \end{array} \]

*116·713. \(\vdash : \text{Hp} *116·712 . \supset . T \in 1 \rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash . *116·712 . *14·21 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : R \in \text{ᗡ}ʻT . \supset . \text{E}! TʻR :\\ [*71·16] &\supset : R \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash . *116·712.711 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : R, S \in \text{ᗡ}ʻT . TʻR = TʻS . \supset . R = S &\qquad \text{(2)}\\ \vdash . (1) . (2) . *71·55 . \supset \vdash . \text{Prop} \end{array} \]

*116.714. \[\begin{align}\vdash : \text{Hp} *116·712 . \mu \in \text{Cl}ʻ\alpha . &R = \hat{\gamma} \hat{x} \{\gamma = \Lambda . x \in \mu . \lor . \gamma = \text{V} . x \in \alpha - \mu\} . \supset .\\ &R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha . \mu = TʻR\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *21·33 . *33·13 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \gamma \in \text{D}ʻR . \supset . \gamma \in \iota ʻ\Lambda \cup \iota ʻ\text{V} &\qquad \text{(1)}\\ \vdash . *21·33 . *33·131 . &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp x \in \text{ᗡ}ʻR . \equiv :\\ &(\exists \gamma ) : \gamma = \Lambda . x \in \mu . \lor . \gamma = \text{V} . x \in \alpha - \mu :\\ [*10·42 . *13·19] &\equiv : x \in \mu . \lor . x \in \alpha - \mu : &\qquad \text{(2)}\\ [*24·411 . \text{Hp}] &\equiv : x \in \alpha &\qquad \text{(3)}\\ \vdash .*21·33.*30·3.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \mu .\supset _{x}.Rʻx=\Lambda :x\in \alpha -\mu .\supset _{x}.Rʻx =\text{V}:\\ [(2).*14·21] &\supset :x\in \text{ᗡ}ʻR.\supset _{x}.\text{E}!Rʻx:\\ [*71·16] &\supset :R\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ \vdash .*21·33. &\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \gamma =\Lambda .\supset :\gamma Rx.\equiv _{x}.x\in \mu :\\ [*32·181] &\supset :\overleftarrow{R}ʻ\gamma =\mu &\qquad \text{(5)}\\ \vdash .(1).(3).(4).*116·12.&\supset \vdash :\text{Hp}.\supset .R\in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha\}_{\Delta }ʻ\alpha &\qquad \text{(6)}\\ \vdash .(5).(6).*116·712. & \supset \vdash :\text{Hp}.\supset .\mu =TʻR &\qquad \text{(7)}\\ \vdash .(6).(7).\supset \vdash .\text{Prop} \end{array} \]

[Pg 169]

*116·715. \(\vdash :\text{Hp}*116·712.\supset .\text{D}ʻT=\text{Cl}ʻ\alpha \)

Dem. \[ \begin{array}{l} \vdash .*116·714.*33·43.&\supset \vdash :\text{Hp}.\supset .\text{Cl}ʻ\alpha \subset \text{D}ʻT &\qquad \text{(1)}\\ \vdash .*21·33. *33·13.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\mu \in \text{D}ʻT.\supset .(\exists R).R\in {(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha }_{\Delta }ʻ\alpha .\mu =\overleftarrow{R}ʻ\Lambda .\\ [*33·151] &\supset .(\exists R).R\in {(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha }_{\Delta }ʻ\alpha .\mu \subset \text{ᗡ}ʻR.\\ [*80·14] &\supset .\mu \subset \alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash . \text{Prop} \end{array} \]

*116·72. \(\vdash .\text{Nc}ʻ\text{Cl}ʻ\alpha =2^{\text{Nc}ʻ\alpha}\)

Dem. \[ \begin{array}{l} \vdash .*116·712·713·715.\supset \vdash .\text{Cl}ʻ\alpha \text{ sm }\{(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha\}_{\Delta }ʻ\alpha &\qquad \text{(1)}\\ \vdash .(1).*116·7.\supset \vdash . \text{Prop} \end{array} \]

*116·8. \(\vdash .\text{Rl}ʻ(\rho \uparrow \sigma )=\dot{s} ʻʻ\text{Cl}ʻ(\sigma \times \rho )\)

Dem. \[ \begin{array}{l} \vdash .*60·2.&\supset \vdash \colon\ldotp R\in \dot{s} ʻʻ\text{Cl}ʻ(\sigma x\rho ).\equiv :(\exists \lambda ).\lambda \subset \sigma \times \rho .R=\dot{s} ʻ\lambda :\\ [*113·101]&\equiv :(\exists \lambda ):P\in \lambda .\supset _{P}.(\exists x,y).x\in \rho .y\in \sigma .P=x\downarrow y:R=\dot{s} ʻ\lambda :\\ [*41·11]&\equiv :(\exists \lambda ):P\in \lambda .\supset _{P}.(\exists x,y).x\in \rho .y\in \sigma .P=x\downarrow y:\\ uRv.\equiv _{u,v}.(\exists P).P\in \lambda .uPv:\\ [*10·56] &\supset :uRv.\supset _{u,v}.(\exists x,y).x\in \rho .y\in \sigma .u(x\downarrow y)v:\\ [*55·13] &\supset :uRv.\supset _{u,v}.u\in \rho .v\in \sigma :\\ [*35·103]. &\supset :R\,\unicode{x2abd}\, \rho \uparrow \sigma &\qquad \text{(1)}\\ \vdash .*35·103.*113·101.\supset \\ \vdash :R\,\unicode{x2abd}\, \rho \uparrow \sigma .\lambda& =\hat{P} \{(\exists x,y).xRy.P=x\downarrow y\}.\supset .\lambda \subset \sigma \times \rho &\qquad \text{(2)}\\ \vdash .*41·11.*13·195.\supset \\ \vdash \colon\ldotp \text{Hp}(2).\supset :u(\dot{s} ʻ\lambda )v.&\equiv .(\exists x,y).xRy.u(x\downarrow y)v.\\ [*55·13] &\equiv .uRv &\qquad \text{(3)}\\ \vdash .(2).(3).&\supset \vdash :R\,\unicode{x2abd}\, \rho \uparrow \sigma .\supset .(\exists \lambda ).\lambda \subset \sigma \times \rho .R=\dot{s} ʻ\lambda &\qquad \text{(4)}\\ \vdash .(1).(4).&\supset \vdash .\text{Prop} \end{array} \]

[Pg 170]

*116·81. \(\vdash .\dot{s} \upharpoonright \text{Cl}ʻ(\sigma \times \rho )\in 1\rightarrow 1\)

Dem. \[ \begin{array}{l} \vdash .*41·13.&\supset \vdash \colon\ldotp \alpha ,\beta \in \text{Cl}ʻ(\sigma \times \rho ).\dot{s} ʻ\alpha =\dot{s} ʻ\beta .x\downarrow y\in \alpha .\supset :x\downarrow y\,\unicode{x2abd}\, \dot{s} ʻ\beta :\\ [*41·11] &\supset :(\exists P).P\in \beta .x\downarrow y\,\unicode{x2abd}\, P:\\ [*113·101.\text{Hp}] &\supset :(\exists P,u,v).P\in \beta .P=u\downarrow v.x\downarrow y\,\unicode{x2abd}\, u\downarrow v:\\ [*55·134·34] &\supset :(\exists P,u,v).P\in \beta .P=u\downarrow v.x\downarrow y=u\downarrow v:\\ [*13·172·13] &\supset :x\downarrow y\in \beta &\qquad \text{(1)}\\ \vdash .(1)\frac{\beta,\,\alpha}{\alpha,\,\beta}.&\supset \vdash :\alpha ,\beta \in \text{Cl}ʻ(\sigma \times \rho ).\dot{s} ʻ\alpha =\dot{s} ʻ\beta .x\downarrow y\in \beta .\supset .x\downarrow y\in \alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*113·101.&\supset \vdash :\alpha ,\beta \in \text{Cl}ʻ(\sigma \times \rho ).\dot{s} ʻ\alpha =\dot{s} ʻ\beta .\supset .\alpha =\beta &\qquad \text{(3)}\\ \vdash .(3).*71·55.*72·163.\supset \vdash .\text{Prop} \end{array} \]

*116·82. \(\vdash .\text{Rl}ʻ(\rho \uparrow \sigma )\text{ sm } \text{Cl}ʻ(\sigma \times \rho ) \quad[*116·8·81]\)

*116·83. \(\vdash .\text{Nc}ʻ\text{Rl}ʻ(\rho \uparrow \sigma )=2^{\text{Nc}ʻ\rho \times _{c}\text{Nc}ʻ\sigma } \quad[*116·82·72 .*113·25]\)

*116·9. \(\vdash :\text{Nc}ʻtʻx=\mu .\supset .\text{Nc}ʻt^{2}ʻx=2^\mu \quad[*116·72.*63·66]\)

*116·901. \(\vdash :\text{Nc}ʻt_{0}ʻ\alpha =\mu .\supset .\text{Nc}ʻtʻ\alpha =2^\mu \quad[*116·72.*63·65]\)

*116·91. \(\vdash :\text{Nc}ʻt_{0}ʻ\alpha =\mu .\supset .\text{Nc}ʻt_{00}ʻ\alpha =2^{\mu ^{2}} \quad[*116·83.*64·5·11]\)

*116·92. \[\begin{align}&\vdash :\text{Nc}ʻt_{0}ʻ\alpha =\mu .\supset .\text{Nc}ʻt_{0}^{1}ʻ\alpha =2^{\mu \times _{c}2^\mu }.\text{Nc}ʻt^{11}ʻ\alpha =2^{2^\mu \times _{c}2^\mu }.\text{etc}.\\ &[*116·83.*64·16.*116·901]\end{align}\]

Summary of *117.

A cardinal \(\mu\) is said to be greater than another cardinal \(\nu\) when there is a class \(\alpha\) which has \(\mu\) terms and has a part which has \(\nu\) terms, while there is no class \(\beta\) which has \(\nu\) terms and has a part which has \(\mu\) terms. The relation "greater than" is transitive and asymmetrical; and by the Schröder-Bernstein theorem, if \(\mu\) is greater than or equal to \(\nu\), and \(\nu\) is greater than or equal to \(\mu\), then \(\mu = \nu\). But we cannot prove that of any two cardinals one must be the greater, unless we assume the multiplicative axiom. The proof then follows from Zermelo's theorem that on that assumption every class can be well-ordered. This subject will be dealt with at a later stage.

The form of the definitions is so arranged as to allow of the inequality of two cardinals in different types. The relevant considerations are the same as for the definitions of addition, multiplication and exponentiation.

Our definition of "\(\mu \gt \nu\)" is

*117·01. \[\begin{align}\mu \gt \nu .=.(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =&\text{N}_{0}\text{c}ʻ\beta .\\ &\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \quad\text{Df}\end{align}\]

We also define "\(\mu \gt \text{Nc}ʻ\alpha\)" as meaning "\(\mu \gt \text{N}_{0}\text{c}ʻ\alpha\)," and "\(\text{Nc}ʻ\alpha \gt \nu\)" as meaning "\(\text{N}_{0}\text{c}ʻ\alpha \gt \nu\)," for the reasons explained in *110. It then easily follows that if \(\mu \gt \nu\), \(\mu\) and \(\nu\) must be homogeneous cardinals (this is part of *117·15); that if \(\mu\) and \(\nu\) are homogeneous cardinals, and \(\mu \gt \nu\), the same holds if we substitute \(\text{ sm }ʻʻ\mu\) and \(\text{ sm }ʻʻ\nu\) for one or both of \(\mu\) and \(\nu\) (*117·16); that

*117·13. \(\vdash :\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha\)

and that

*117·14. \(\vdash :\mu \gt \nu .\equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta\)

We cannot define "\(\mu \geq \nu\)" as "\(\mu \gt \nu .\lor.\mu =\nu\)," because "\(\mu =\nu\)" restricts \(\mu\) and \(\nu\) too much by requiring that they should be of the same type, and restricts them too little by not requiring that they should both be existent cardinals. To avoid both these inconveniences, we put

*117·05. \(\mu \geq \nu .=:\mu \gt \nu .\lor.\mu ,\nu \in \text{N}_{0}\text{C}.\mu =\text{ sm }ʻʻ\nu \quad\text{Df}\)

The use of this definition is chiefly through the propositions

*117·108. \(\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv :\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta .\lor.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)

[Pg 172]

*117·24. \(\vdash :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)

In *117·2, we repeat the Schröder-Bernstein theorem (*73·88), which is required in most of the remaining propositions of this number. It leads at once to the propositions

*117·22. \(\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)

(which practically supersedes the definition of "\(\geq\)")

*117·221. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta\)

*117·222. \(\vdash :\beta \subset \alpha .\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)

*117·23. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)

This last proposition may be called the Schröder-Bernstein theorem with as much propriety as *73·88; the two are scarcely different.

If we now revert to the definition of \(\mu \gt \nu\), or to *117·13, and apply *117·22, we see (*117·26) that "\(\text{Nc}ʻ\alpha>\text{Nc}ʻ\beta\)" may be conveniently regarded as asserting \(\text{Nc}ʻ\alpha \geq\text{Nc}ʻ\beta .{\sim}(\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha)\); in fact, the best ideas to work with are \(\geq\) and its converse \(\leq\), which for practical purposes we regard as defined by *117·22, and from which we derive \(\gt\) and \(\lt\) . The relation \(\gt\) will be the product of \(\geq\) into the negation of its converse; this holds for \(\mu\) and \(\nu\) (*117·281) as well as for \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\).

*117·3·31 constitute an important use of *110·72, namely to prove that one existent cardinal is greater than another or equal to it when the first can be obtained by adding to the second (where what is added must be a cardinal). That is to say, we have

*117·3. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \varpi ).\varpi \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi\)

*117·31. \(\vdash \colon\ldotp \mu \geq \nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \varpi ).\varpi \in \text{NC}.\mu =\nu +_{c}\varpi\)

*117·4—·471 are concerned in proving that \(\gt\) and \(\geq\) are transitive, that \(\gt\) is asymmetrical (*117·42), and allied propositions.

Our next set of propositions is concerned with 0 and 1 and 2. We prove that a homogeneous cardinal is whatever is greater than or equal to 0 (*117·501); that a homogeneous cardinal other than 0 is whatever is greater than 0 (*117·511); that a homogeneous cardinal other than 0 is whatever is greater than or equal to 1 (*117·531); and that a homogeneous cardinal other than 0 and 1 is whatever is greater than 1 (*117·55), and is whatever is greater than or equal to 2 (*117·551).

We next prove a set of propositions concerning \(\geq\) which have no analogues for \(\gt\), except when the cardinals concerned are finite. Thus e.g. we prove

*117·561. \(\vdash :\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}.\supset .\mu +_{c}\varpi \geq \nu +_{c}\varpi\)

If we substitute \(\gt\) for \(\geq\), this no longer holds. Thus e.g. put \(\mu =2\), \(\nu =1\), \(\varpi =\aleph _{0}\) (cf. *123); then \(\mu >\nu\), but \(\mu +_{c}\varpi =\nu +_{c}\varpi =\varpi\). Similar remarks apply to the analogous propositions (*117·571·581·591) on multiplication and exponentiation.

[Pg 173]

We prove next that a sum is greater than or equal to either of its summands (*117·6); that a product neither of whose factors vanishes is greater than or equal to either of its factors (*117·62); that, assuming \(\mu\) and \(\nu\) are existent cardinals, then if they are neither 0 nor 1, their product is greater than or equal to their sum (*117·631), and if \(\mu\) is neither 0 nor 1, then \(\mu ^\nu \geq \mu\times _{c} \nu\) (*117·652).

The last important proposition in this number is Cantor's theorem

*117·661. \(\vdash : \mu \in \text{N}_{0}\text{C} . \supset . 2^\mu > \mu\)

which follows immediately from *102·72 and *116·72.

The propositions of this number are much used in the following section, on finite and infinite.

*117·01. \[\begin{align}\mu \gt \nu . = . (\exists \alpha , \beta ) . &\mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta .\\ &\exists ! \text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta . {\sim} \exists ! \text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \quad\text{Df}\end{align}\]

*117·02. \(\mu \gt \text{Nc}ʻ\alpha . = . \mu \gt \text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)

*117·03. \(\text{Nc}ʻ\alpha \gt \nu . = . \text{N}_{0}\text{c}ʻ\alpha \gt \nu \quad\text{Df}\)

*117·04. \(\mu \lt \nu . = . \nu \gt \mu \quad\text{Df}\)

*117·05. \(\mu \geq \nu . = : \mu \gt \nu . \lor . \mu , \nu \in \text{N}_{0}\text{C} .\mu = \text{ sm }ʻʻ\nu \quad\text{Df}\)

*117·06. \(\mu \leq \nu . = . \nu \geq \mu \quad\text{Df}\)

The analogues of *117·02·03 are to be applied also to *117·04·05·06.

*117·1. \[\begin{align}\vdash : \mu \gt \nu . &\equiv . (\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta .\\ &\exists ! \text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta . {\sim} \exists ! \text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \quad[(*117·01)]\end{align}\]

*117·101. \(\vdash : \mu \gt \text{Nc}ʻ\beta . \equiv . \mu \gt \text{N}_{0}\text{c}ʻ\beta \quad[(*117·02)]\)

*117·102. \(\vdash : \text{Nc}ʻ\alpha > \nu . \equiv . \text{N}_{0}\text{c}ʻ\alpha > \nu \quad[(*117·03)]\)

*117·103. \(\vdash : \mu < \nu . \equiv . \nu > \mu \quad[(*117·04)]\)

*117·104. \(\vdash \colon\ldotp \mu \geq \nu . \equiv : \mu > \nu . \lor . \mu , \nu \in \text{N}_{0}\text{C} . \mu = \text{ sm }ʻʻ\nu \quad[(*117·05)]\)

*117·105. \(\vdash : \mu \leq \nu . \equiv . \nu \geq \mu \quad[(*117·06)]\)

*117·106. \(\vdash : \text{Nc}ʻ\alpha > \text{Nc}ʻ\beta . \equiv . \text{N}_{0}\text{c}ʻ\alpha > \text{N}_{0}\text{c}ʻ\beta \quad [*117·101·102]\)

*117·107. \(\vdash : \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \equiv . \text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash . *117·104·106 . \supset \\ \vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . &\equiv : \text{N}_{0}\text{c}ʻ\alpha > \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\alpha , \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{Nc}ʻ\alpha = \text{ sm }ʻʻ\text{Nc}ʻ\beta :\\ [*100·511 . *103·22] &\equiv : \text{N}_{0}\text{c}ʻ\alpha > \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\alpha , \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta :\\ [*103·16] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\alpha , \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{Nc}ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta :\\ [*103·21] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{Nc}ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta :\\ [*103·16] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta :\\ [*103·2] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta :\\ [*103·4] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{N}_{0}\text{c}ʻ\alpha = \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\beta :\\ [*103·21 . *117·104] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

[Pg 174]

*117·108. \[\begin{align}&\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv :\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta .\lor.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \\ &[*117·107·106·104. *103·16·4]\end{align}\]

*117·11. \(\vdash \colon\ldotp \alpha \text{ sm } \alpha'.\beta \text{ sm } \beta'. \supset : \exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta'\)

Dem. \[ \begin{array}{l} \vdash .*100·321.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta' &\qquad \text{(1)}\\ \vdash .*73·21.\supset \\ \vdash : R\in 1\rightarrow &1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=\alpha'.\gamma \subset \alpha .\gamma \in \text{Nc}ʻ\beta .\supset .\\ &\breve{R} ʻʻ\gamma \subset \alpha'.\breve{R} ʻʻ\gamma \in \text{Nc}ʻ\beta .\\ [*60·2]&\supset .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta &\qquad \text{(2)}\\ \vdash .(2).*10·11·23·35.*73·1.\supset \\ \vdash :\alpha \text{ sm } \alpha'.\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .&\supset .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta &\qquad \text{(3)}\\ \vdash .(3)\frac{\alpha',\,\alpha}{\alpha,\,\alpha'}.&\supset \vdash :\alpha \text{ sm } \alpha'.\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta .\supset .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta &\qquad \text{(4)}\\ \vdash .(3).(4).&\supset \vdash \colon\ldotp \alpha \text{ sm } \alpha '.\supset :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha ʻ\cap \text{Nc}ʻ\beta &\qquad \text{(5)}\\ \vdash .(1).(5).&\supset \vdash . \text{Prop} \end{array} \]

*117·12. \[\begin{align}\vdash \colon\ldotp \mu >&\nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\\ &\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*117·1·11.\supset \\ \vdash \colon\ldotp \mu >\nu .\equiv :(\exists \alpha ,\beta ):&\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ &\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ [*103·12]&\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ &{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ &\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ [*10·55] &\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\alpha \in \mu .\beta \in \nu :\\ &\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ [*103·12·2]&\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma \colon\ldotp \\ &\supset \vdash . \text{Prop} \end{array} \]

*117·121. \[\begin{align}\vdash \colon\ldotp \mu >&\nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\\ &\alpha \in \mu .\supset _{\alpha }.(\exists \beta ).\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*117·1·11.\supset \\ \vdash \colon\ldotp \mu >\nu .\equiv :&(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ &{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ &\gamma \in \mu .\supset _{\gamma }.(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma \\ [*103·12.*10·55]&\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\alpha \in \mu :\\ &\gamma \in \mu .\supset _{\gamma }.(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}^\gamma \\ [*103·12·2] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\gamma \in \mu .\supset _{\gamma }.(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .\\ &{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

[Pg 175]

The above proof is given shortly because it proceeds on the same lines as *117·12. In applying *10·55, the \(\phi x\) of that proposition is replaced by \(\alpha \in \mu\), and the \(\psi x\) is replaced by \[ (\exists \beta ).\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha . \]

*117·13. \(\vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*117·106.\supset \\ \vdash \colon\ldotp \text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .&\equiv :\text{N}_{0}\text{c}ʻ\alpha >\text{N}_{0}\text{c}ʻ\beta :\\ [*103·2.*117·12] &\equiv :\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\delta \in \text{N}_{0}\text{c}ʻ\beta .\supset _{\gamma ,\delta }.\\ &\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ [*100·31.*117·11] &\equiv :\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\delta \in \text{N}_{0}\text{c}ʻ\beta .\supset _{\gamma ,\delta }.\\ &\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ [*10·23] &\equiv :\exists !\text{N}_{0}\text{c}ʻ\alpha .\exists !\text{N}_{0}\text{c}ʻ\beta .\supset .\\ &\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ [*103·13] &\equiv :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

*117·14. \[\begin{align}&\vdash :\mu \gt \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta \\ &[*117·1·13]\end{align}\]

*117·15. \(\vdash :\mu \gt \nu .\equiv .\mu ,\nu \in \text{N}_{0}\text{C}.\exists !sʻ\text{Cl}ʻʻ\mu \cap \text{ sm }ʻʻ\nu .{\sim}\exists !sʻ\text{Cl}ʻʻ\nu \cap \text{ sm }ʻʻ\mu\)

Dem. \[ \begin{array}{l} \vdash .*103·4.*117·1.\supset \\ \vdash \colon\ldotp \mu >\nu .&\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu .\\ &{\sim}\exists !\text{Cl}ʻ\beta \cap \text{ sm }ʻʻ\mu :\\ [*103·2·26] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu .\\ &{\sim}\exists !\text{Cl}ʻ\beta \cap \text{ sm }ʻʻ\mu :\\ [*117·11] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu :\\ &\delta \in \nu .\supset _{\delta }.{\sim}\exists !\text{Cl}ʻ\delta \cap \text{ sm }ʻʻ\mu :\\ [*103·13.*10·51] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \alpha ).\alpha \in \mu .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu :\\ &{\sim}(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\delta \cap \text{ sm }ʻʻ\mu :\\ [*40·4.*60·2] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}.\exists !sʻ\text{Cl}ʻʻ\mu \cap \text{ sm }ʻʻ\nu .\\ &{\sim}\exists !sʻ\text{Cl}ʻʻ\nu \cap \text{ sm }ʻʻ\mu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

The advantage of this proposition is that it expresses "\(\mu \gt \nu\)" in terms of \(\mu\) and \(\nu\) alone, without the auxiliary \(\alpha\) and \(\beta\) of the definition.

*117·16. \[\begin{align}&\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\supset :\mu \gt \nu .\equiv .\text{ sm }ʻʻ\mu \gt \nu .\equiv .\mu >\text{ sm }ʻʻ\nu .\equiv .\text{ sm }ʻʻ\mu >\text{ sm }ʻʻ\nu \\ &[*117·14.*103·4]\end{align}\]

*117·2. \(\vdash :\alpha \text{ sm } \alpha ʻ.\beta \text{ sm } \beta ʻ. \beta ʻ\subset \alpha .\alpha ʻ\subset \beta .\supset .\alpha \text{ sm } \beta \quad[*73·88]\)

[Pg 176]

This proposition (which is the Schröder-Bernstein theorem) is fundamental in the theory of greater and less.

*117·21. \[\begin{align}&\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \\ &[*117·2.*100·321]\end{align}\]

*117·211. \(\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*100·3.*60·34.\supset \vdash :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\supset .\alpha \in \text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\beta \in \text{Cl}ʻ\beta\\ \cap \text{Nc}ʻ\alpha &\quad \text{(1)}\\ \vdash .(1).*117·21.\supset \vdash .\text{Prop} \end{array} \]

*117·22. \(\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*117·13. &\supset \vdash :\text{Hp}.{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta &\qquad \text{(1)}\\ \vdash .*117·211.&\supset \vdash :\text{Hp}.\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).*117·108.\supset \vdash .\text{Prop} \end{array} \]

*117·221. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta \quad[*117·22.*60·2.*100·1]\)

*117·222. \(\vdash :\beta \subset \alpha .\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta \quad[*117·221]\)

*117·23. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \quad[*117·211·22]\)

*117·24. \(\vdash :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*117·104·14.\supset \vdash \colon\ldotp \mu \geq \nu .&\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\lor.\\ &(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu =\text{ sm }ʻʻ\nu \\ [*103·4.*13·193] &\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\lor.\\ &(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\beta :\\ [*103·16] &\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\lor.\\ &(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta :\\ [*11·41.*117·108] &\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta \colon\ldotp \\ &\supset \vdash .\text{Prop} \end{array} \]

*117·241. \[\begin{align}&\vdash :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta \\ &[*117·24·22]\end{align}\]

*117·242. \[\begin{align}&\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta \\ &[*117·241.*103·26]\end{align}\]

*117·243. \[\begin{align}&\vdash \colon\ldotp \mu \geq \nu .\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta :(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta \\ &[*117·24·221]\end{align}\]

*117·244. \[\begin{align}\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\supset :\mu \geq \nu .\equiv .&\text{ sm }ʻʻ\mu \geq \nu .\equiv .\mu \geq \text{ sm }ʻʻ\nu .\equiv .\\ &\text{ sm }ʻʻ\mu \geq \text{ sm }ʻʻ\nu \quad[*117·24.*103·4]\end{align}\]

[Pg 177]

*117·25. \(\vdash :\mu \geq \nu .\nu \geq \mu .\equiv .\mu ,\nu \in \text{N}_{0}\text{C}.\text{ sm }ʻʻ\mu =\text{ sm }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash .*117·24.\supset \\ \vdash :\mu \geq \nu .\nu \geq \mu .&\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\delta .\\ &\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\delta \geq \text{Nc}ʻ\gamma .\\ [*117·107] &\equiv .(\exists \alpha ,\beta ,\gamma,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\delta .\\ &\text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\delta \geq \text{N}_{0}\text{c}ʻ\gamma .\\ [*13·193] &\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\delta .\\ &\text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\beta \geq \text{N}_{0}\text{c}ʻ\alpha .\\ [*117·107·23] &\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\delta .\\ &\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\\ [*11·45.*103·2] &\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu ,\nu \in \text{N}_{0}\text{C}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\\ [*103·4] &\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu ,\nu \in \text{N}_{0}\text{C}.\text{ sm }ʻʻ\mu =\text{ sm }ʻʻ\nu .\\ [*11·45.*103·2] &\equiv .\mu ,\nu \in \text{N}_{0}\text{C}.\text{ sm }ʻʻ\mu =\text{ sm }ʻʻ\nu :\supset \vdash .\text{Prop} \end{array} \]

*117·26. \(\vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*117·13.*13·12.\text{Transp}.&\supset \vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\supset .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta :\\ [*117·108] &\supset \vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta &\qquad \text{(1)}\\ \vdash .*117·108.*5·6.&\supset \vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta .\supset .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*117·27. \[\begin{align}&\vdash :\text{Nc}ʻ\alpha <\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta \\ &[*117·26·103·105]\end{align}\]

*117·28. \[\begin{align}&\vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .{\sim}(\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha )\\ &[*117·22·13]\end{align}\]

*117·2. \(\vdash :\mu >\nu .\equiv .\mu \geq \nu .{\sim}(\nu \geq \mu ) \quad[*117·14·28·24]\)

*117·29. \(\vdash :\text{Nc}ʻ\alpha <\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .{\sim}(\text{Nc}ʻ\beta \leq \text{Nc}ʻ\alpha ) \quad[*117·28]\)

*117·291. \(\vdash :\mu <\nu .\equiv .\mu \leq \nu .{\sim}(\nu \leq \mu ) \quad[*117·281]\)

*117·3. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \varpi ).\varpi \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi\)

Dem. \[ \begin{array}{l} \vdash .*117·221.\supset \vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta &.\equiv .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha .\\ [*110·72] &\equiv .(\exists \varpi ).\varpi \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi :\\ &\supset \vdash .\text{Prop} \end{array} \]

[Pg 178]

*117·31. \(\vdash \colon\ldotp \mu \geq \nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \varpi ).\varpi \in \text{NC}.\mu =\nu +_{c}\varpi\)

Dem. \[ \begin{array}{l} \vdash .*117·24·3.\supset \\ \vdash \colon\ldotp \mu \geq \nu .&\equiv :(\exists \alpha ,\beta ,\varpi ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi :\\ [(*110·03)] &\equiv :(\exists \alpha ,\beta ,\varpi ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha =\nu +_{c}\varpi :\\ [*103·16.*110·42] &\equiv :(\exists \alpha ,\beta ,\varpi ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu =\nu +_{c}\varpi :\\ [*103·2] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \varpi ).\mu =\nu +_{c}\varpi \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*117·32. \(\vdash :\mu \geq \nu .\exists !\text{ sm }ʻʻ\mu \cap tʻ\alpha .\supset .\exists !\text{ sm }ʻʻ\nu \cap tʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*117·241.*103·4.\supset \\ \vdash :\text{Hp}.\supset .(\exists \beta ,\gamma ).\mu =\text{N}_{0}\text{c}ʻ\beta .\nu =\text{N}_{0}\text{c}ʻ\gamma .\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\gamma .\text{ sm }ʻʻ\mu =\text{Nc}ʻ\beta .\\ \text{ sm }ʻʻ\nu =\text{Nc}ʻ\gamma &\qquad \text{(1)}\\ \vdash .*63·105·371.*73·12.\supset \\ \vdash :R\in \rho \,\overline{\text{ sm }}\, \beta .\rho \in tʻ\alpha .\sigma \subset \beta .\sigma \text{ sm }\gamma .\supset .Rʻʻ\sigma \in tʻ\alpha .Rʻʻ\sigma \text{ sm }\gamma &\qquad \text{(2)}\\ \vdash .(2).*73·04.\supset \vdash :\rho \in \text{Nc}ʻ\beta \cap tʻ\alpha .\sigma \in \text{Cl}ʻ\beta \cap \text{Nc}ʻ\gamma .\supset .\exists !\text{Nc}ʻ\gamma \cap tʻ\alpha &\qquad \text{(3)}\\ \vdash .(1).(3).\supset \vdash . \text{Prop} \end{array} \]

The above proposition shows that if a cardinal \(\mu\) exists in a given type, so do all smaller cardinals.

*117·4. \(\vdash :\mu \geq \nu .\nu \geq \varpi .\supset .\mu \geq \varpi\)

Dem. \[ \begin{array}{l} \vdash .*117·243.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \alpha ,\beta ,\gamma ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\varpi =\text{N}_{0}\text{c}ʻ\gamma :\\ &(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta :(\exists \sigma ).\sigma \subset \beta .\sigma \text{ sm }\gamma :\\ [*117·11.*60·2.*100·1] &\supset :(\exists \alpha ,\beta ,\gamma ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\varpi =\text{N}_{0}\text{c}ʻ\gamma :\\ &(\exists \rho ,\tau ).\rho \subset \alpha .\rho \text{ sm }\beta .\tau \subset \rho .\tau \text{ sm }\gamma :\\ [*22·44] &\supset :(\exists \alpha ,\gamma ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\varpi =\text{N}_{0}\text{c}ʻ\gamma :(\exists \tau ).\tau \subset \alpha .\tau \text{ sm }\gamma :\\ [*117·243] &\supset :\mu \geq \varpi \colon\ldotp \supset \vdash . \text{Prop} \end{array} \]

*117·41. \(\vdash :\mu \leq \nu .\nu \leq \varpi .\supset .\mu \leq \varpi \quad[*117·4]\)

*117·42. \(\vdash :{\sim}(\mu >\mu ).{\sim}(\mu <\mu )\)

Dem. \[ \begin{array}{l} \vdash .*117·15.*13·12.\text{Transp}.\supset \vdash :\mu >\nu .\supset .\mu \neq \nu :\supset \vdash . \text{Prop} \end{array} \]

*117·43. \(\vdash :\mu \geq \nu .{\sim}(\mu \geq \varpi ).\supset .{\sim}(\nu \geq \varpi ) \quad[*117·4.\text{Transp}]\)

*117·44. \(\vdash :\nu \geq \varpi .{\sim}(\mu \geq \varpi ).\supset .{\sim}(\mu \geq \nu ) \quad[*117·4.\text{Transp}]\)

*117·45. \(\vdash :\mu \geq \nu .\nu >\varpi .\supset .\mu >\varpi\)

Dem. \[ \begin{array}{l} \vdash .*117·281.\supset \vdash :\text{Hp}.&\supset .\mu \geq \nu .\nu \geq \varpi .{\sim}(\varpi \geq \nu ).\\ [*117·4·44] &\supset .\mu \geq \varpi .{\sim}(\varpi \geq \mu ).\\ [*117·281] &\supset .\mu >\varpi :\supset \vdash . \text{Prop} \end{array} \]

*117·46. \(\vdash :\mu >\nu .\nu \geq \varpi .\supset .\mu >\varpi \quad[\text{Proof as in *117·45}]\)

*117·47. \(\vdash :\mu >\nu .\nu >\varpi .\supset .\mu >\varpi \quad[*117·45·104]\)

[Pg 179]

*117·471. \(\vdash :\mu <\nu .\nu <\varpi .\supset .\mu <\varpi \quad[*117·47·103]\)

*117·5. \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu \geq 0\)

Dem. \[ \begin{array}{l} \vdash .*60·3.*100·3.&\supset \vdash .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\Lambda .\\ [*117·22] &\supset \vdash .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\Lambda .\\ [*117·107.*101·1]&\supset \vdash .\text{N}_{0}\text{c}ʻ\alpha \geq 0 &\qquad \text{(1)}\\ \vdash .(1).*103·2. &\supset \vdash .\text{Prop} \end{array} \]

*117·501. \(\vdash :\mu \in \text{N}_{0}\text{C}.\equiv .\mu \geq 0 \quad[*117·5·104]\)

*117·51. \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\mu >0\)

Dem. \[ \begin{array}{l} \vdash .*101·15.\supset \vdash :\text{Hp}.\supset .\mu \neq \text{ sm }ʻʻ0 &\qquad \text{(1)}\\ \vdash .(1).*117·5·104.\supset \vdash .\text{Prop} \end{array} \]

*117·511. \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\equiv .\mu >0 \quad[*117·51·15·42]\)

*117·52. \(\vdash :\exists !\xi .\supset .\text{Nc}ʻ\xi \geq 1\)

Dem. \[ \begin{array}{l} \vdash .*51·2.\supset \vdash :\text{Hp}.&\supset .(\exists x).\iota ʻx\subset \xi .\\ [*117·222] &\supset .(\exists x).\text{Nc}ʻ\xi \geq \text{Nc}ʻ\iota ʻx.\\ [*101·2] &\supset .\text{Nc}ʻ\xi \geq 1:\supset \vdash .\text{Prop} \end{array} \]

*117·53. \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\mu \geq 1\)

Dem. \[ \begin{array}{l} \vdash .*101·16.*103·2.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ).\text{N}_{0}\text{c}ʻ\alpha =\mu .\exists !\alpha .\\ [*117·52] &\supset .(\exists \alpha ).\text{N}_{0}\text{c}ʻ\alpha =\mu .\text{Nc}ʻ\alpha \geq 1.\\ [*117·107] &\supset .\mu \geq 1:\supset \vdash .\text{Prop} \end{array} \]

*117·531. \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\equiv .\mu \geq 1\)

Dem. \[ \begin{array}{l} \vdash .*117·104. &\supset \vdash :\mu \geq 1.\supset .\mu \in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ \vdash .*117·51.*101·22.&\supset \vdash .1>0.\\ [*117·45] &\supset \vdash :\mu \geq 1.\supset .\mu >0.\\ [*117·42] &\supset .\mu \neq 0 &\qquad \text{(2)}\\ \vdash .(1).(2).*117·53.&\supset \vdash .\text{Prop} \end{array} \]

*117·54. \(\vdash \colon\ldotp 1\geq \mu .\equiv :\mu =0.\lor.\mu =1\)

Dem. \[ \begin{array}{l} \vdash .*117·241.*101·2.*52·22.\supset \\ \vdash \colon\ldotp 1\geq \mu .&\equiv :(\exists \alpha ,x).\mu =\text{N}_{0}\text{c}ʻ\alpha .\exists !\text{Nc}ʻ\alpha \cap \text{Cl}ʻ\iota ʻx:\\ [*60·362] &\equiv :(\exists \alpha ,x):\mu =\text{N}_{0}\text{c}ʻ\alpha :\exists !\text{Nc}ʻ\alpha \cap \iota ʻ\Lambda .\lor.\exists !\text{Nc}ʻ\alpha \cap \iota ʻ\iota ʻx:\\ [*51·31] &\equiv :(\exists \alpha ,x):\mu =\text{N}_{0}\text{c}ʻ\alpha :\Lambda \in \text{Nc}ʻ\alpha .\lor.\iota ʻx\in \text{Nc}ʻ\alpha :\\ [*101·17·29] &\equiv :(\exists \alpha ,x):\mu =\text{N}_{0}\text{c}ʻ\alpha :\text{Nc}ʻ\alpha =\text{Nc}ʻ\Lambda .\lor.\text{Nc}ʻ\alpha =\text{Nc}ʻ\iota ʻx:\\ [*103·16] &\equiv :(\exists \alpha ,x).\mu =\text{N}_{0}\text{c}ʻ\alpha :\mu =\text{Nc}ʻ\Lambda .\lor.\mu =\text{Nc}ʻ\iota ʻx:\\ [*101·1·2] &\equiv :(\exists \alpha ).\mu =\text{N}_{0}\text{c}ʻ\alpha :\mu =0.\lor.\mu =1:\\ [*103·2·5·51] &\equiv :\mu =0.\lor.\mu =1\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

[Pg 180]

*117·55. \(\vdash : \mu > 1 . \equiv . \mu \in \text{N}_{0}\text{C} - \iota ʻ0 - \iota ʻ1\)

Dem. \[ \begin{array}{l} \vdash . *117·281 . \supset \vdash : \mu > 1 . &\equiv . \mu \geq 1 . {\sim} (1 \geq \mu ) .\\ [*117·531·54] &\equiv . \mu \in \text{N}_{0}\text{C} - \iota ʻ0 . \mu \neq 0 . \mu \neq 1 .\\ [*51·15] &\equiv . \mu \in \text{N}_{0}\text{C} - \iota ʻ0 - \iota ʻ1 : \supset \vdash . \text{Prop} \end{array} \]

*117·551. \[\begin{align}\vdash \colon\ldotp \mu \in \text{N}_{0}\text{C} - \iota ʻ0 - &\iota ʻ1 . \equiv :\\ &(\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y) . x, y \in \alpha . x \neq y : \equiv . \mu \geq 2\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *103·2 . \supset \vdash \colon\ldotp &\mu \in \text{N}_{0}\text{C} - \iota ʻ0 - \iota ʻ1 . \equiv :\\ &(\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \text{N}_{0}\text{c}ʻ\alpha \neq 0 . \text{N}_{0}\text{c}ʻ\alpha \neq 1 :\\ [*101·14] &\equiv : (\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \exists ! \alpha . \text{N}_{0}\text{c}ʻ\alpha \neq 1 :\\ [*103·26] &\equiv : (\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \exists ! \alpha . \alpha {\sim} \in 1 :\\ [*52·41] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y) . x, y \in \alpha . x \neq y : &\qquad \text{(1)}\\ [*54·26 . *51·2] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y) . \iota ʻx \cup \iota ʻy \subset \alpha . \iota ʻx \cup \iota ʻy \in 2 :\\ [*13·195] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y, \beta ) . \beta = \iota ʻx \cup \iota ʻy . \beta \subset \alpha . \beta \in 2 :\\ [*54·101] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists \beta ) . \beta \subset \alpha . \beta \in 2 :\\ [*117·241] &\equiv : \mu \geq 2 &\qquad \text{(2)}\\ \vdash . (1) . (2) . \supset \vdash . \text{Prop} \end{array} \]

*117·56. \(\vdash : \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \supset . \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\gamma \geq \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *110·12 . *117·221 . \supset \\ \vdash : \text{Hp} . &\supset . (\exists \delta ) . \delta \subset \downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\alpha . \delta \text{ sm } \downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta .\\ [*110·11 . *73·71 . (*110·01)] &\supset . (\exists \delta ) . \delta \cup \Lambda _{\alpha } \downarrow ʻʻ\iotaʻʻ\gamma \subset \alpha + \gamma .\\ &\delta \cup \Lambda _{\alpha } \downarrow ʻʻ\iotaʻʻ\gamma \text{ sm } (\beta + \gamma ) .\\ [*117·221] &\supset . \text{Nc}ʻ(\alpha + \gamma ) \geq \text{Nc}ʻ(\beta + \gamma ) .\\ [*110·3] &\supset . \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\gamma \geq \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma : \supset \vdash . \text{Prop} \end{array} \]

*117·561. \(\vdash : \mu \geq \nu . \varpi \in \text{N}_{0}\text{C} . \supset . \mu +_{c} \varpi \geq \nu +_{c} \varpi \quad[*117·56]\)

The proof of *117·561 follows from *117·56 in the same way as the proof of *117·31 follows from *117·3. In the remainder of this number we shall omit proofs of this kind.

*117·57. \(\vdash : \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \supset . \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\gamma \geq \text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash . *37·2 . &\supset \vdash : \rho \subset \alpha . \supset . \gamma \downarrow_{,,}ʻʻ\rho \subset \gamma \downarrow_{,,}ʻʻ\alpha .\\ [*40·161 . *113·1] &\supset . \rho \times \gamma \subset \alpha \times \gamma &\qquad \text{(1)}\\ \vdash . *113·13 . &\supset \vdash \rho \text{ sm } \beta . \supset . \rho \times \gamma \text{ sm } \beta \times \gamma &\qquad \text{(2)}\\ \vdash . (1) . (2) . &\supset \vdash : \rho \subset \alpha . \rho \text{ sm } \beta . \supset . \rho \times \gamma \subset \alpha \times \gamma . \rho \times \gamma \text{ sm } \beta \times \gamma .\\ [*117·221] &\supset . \text{Nc}ʻ(\alpha \times \gamma ) \geq \text{Nc}ʻ(\beta \times \gamma ) &\qquad \text{(3)}\\ \vdash . (3) . *117·221 . \supset \vdash . \text{Prop} \end{array} \]

[Pg 181]

*117·571. \(\vdash .\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}.\supset .\mu \times _{c}\varpi \geq \nu \times _{c}\varpi \quad[*117·57]\)

*117·58. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\supset .(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\gamma }\geq (\text{Nc}ʻ\beta )^{\text{Nc}ʻ\gamma }\)

Dem. \[ \begin{array}{l} \vdash .*35·432·82.\supset \vdash : \rho \subset \alpha .&\supset .\rho \uparrow \gamma \,\unicode{x2abd}\, \alpha \uparrow \gamma .\\ [*80·15] &\supset .(\rho \uparrow \gamma )_{\Delta }ʻ\gamma \subset (\alpha \uparrow \gamma )_{\Delta }ʻ\gamma &\qquad \text{(1)}\\ \vdash .*116·15·19.&\supset \vdash :\rho \text{ sm }\beta .\supset .(\rho \uparrow \gamma )_{\Delta }ʻ\gamma \text{ sm }(\beta \uparrow \gamma )_{\Delta }ʻ\gamma &\qquad \text{(2)}\\ \vdash .(1).(2).*117·221.\supset \\ \vdash :\rho \subset \alpha .\rho \text{ sm }\beta .&\supset .\text{Nc}ʻ(\alpha \uparrow \gamma )_{\Delta }ʻ\gamma \geq \text{Nc}ʻ(\beta \uparrow \gamma )_{\Delta }ʻ\gamma .\\ [*116·15·25] &\supset .(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\gamma }\geq (\text{Nc}ʻ\beta )^{\text{Nc}ʻ\gamma } &\qquad \text{(3)}\\ \vdash .(3).*117·221.&\supset \vdash . \text{Prop} \end{array} \]

*117·581. \(\vdash :\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}.\supset .\mu ^{\varpi }\geq \nu ^{\varpi } \quad[*117·58]\)

The two following propositions are lemmas for *117·59.

*117·582. \[\begin{align}\vdash :\exists !\gamma .\beta \subset \alpha .\sigma \in \gamma \,\,\text{exp}\,\,(\alpha -\beta ).\supset .&(\cup \sigma )\upharpoonright (\gamma \,\,\text{exp}\,\,\beta )\in 1\rightarrow 1.\\ &(\cup \sigma )ʻʻ(\gamma \,\,\text{exp}\,\,\beta )\subset \gamma \,\,\text{exp}\,\,\alpha \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*116·183.&\supset \vdash :\rho \in (\gamma \,\,\text{exp}\,\,\beta ).\sigma \in \gamma \,\,\text{exp}\,\,(\alpha -\beta ).\supset .\rho \subset \beta \times \gamma .\sigma \subset (\alpha -\beta )\times \gamma .\\ [*113·19.*24·21] &\supset .\rho \cap \sigma =\Lambda &\qquad \text{(1)}\\ \vdash .(1).*24·481.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \rho ,\rho ʻ\in (\gamma \,\,\text{exp}\,\,\beta ).\supset :\rho \cup \sigma =\rho ʻ\cup \sigma .\equiv .\rho =\rho ʻ\colon\ldotp \\ [*71·58] &\supset \colon\ldotp (\cup \sigma )\upharpoonright (\gamma \,\,\text{exp}\,\,\beta )\in 1\rightarrow 1 &\qquad \text{(2)}\\ \vdash .*113·191.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \downarrow_{,,}ʻʻ\beta \cap \gamma \downarrow_{,,}ʻʻ(\alpha -\beta )=\Lambda :\\ [*115·14.(*116·01)] &\supset :\rho \in (\gamma \,\,\text{exp}\,\,\beta ).\supset .\rho \cup \sigma \in \text{Prod}ʻ\{\gamma \downarrow_{,,}ʻʻ\beta \cup \gamma \downarrow_{,,}ʻʻ(\alpha -\beta )\}.\\ [*37·22.*24·411] &\supset .\rho \cup \sigma \in (\gamma \,\,\text{exp}\,\,\alpha ):\\ [*37·61] & \supset :(\cup \sigma )ʻʻ(\gamma \,\,\text{exp}\,\,\beta )\subset \gamma \,\,\text{exp}\,\,\alpha &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*117·583. \(\vdash :\beta \subset \alpha .\exists !\gamma .\supset .(\exists \tau ).\tau \subset \gamma \,\,\text{exp}\,\,\alpha .\tau \text{ sm }(\gamma \,\,\text{exp}\,\,\beta )\)

Dem. \[ \begin{array}{l} \vdash .*116·171.\supset \vdash :\text{Hp}.\supset .\exists !\gamma \,\,\text{exp}\,\,(\alpha -\beta ) &\qquad \text{(1)}\\ \vdash .(1).*117·582.*73·15.\supset \vdash .\text{Prop} \end{array} \]

*117·59. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\exists !\gamma .\supset .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\alpha }\geq (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\beta }\)

Dem. \[ \begin{array}{l} \vdash .*117·221.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta :\exists !\gamma :\\ [*117·583] &\supset :(\exists \rho ,\tau ).\rho \subset \alpha .\rho \text{ sm }\beta .\tau \subset \gamma \,\,\text{exp}\,\,\alpha .\tau \text{ sm }(\gamma \,\,\text{exp}\,\,\rho ):\\ [*116·19] &\supset :(\exists \tau ).\tau \subset \gamma \,\,\text{exp}\,\,\alpha .\tau \text{ sm }(\gamma \,\,\text{exp}\,\,\beta ):\\ [*117·221] &\supset :\text{Nc}ʻ(\gamma \,\,\text{exp}\,\,\alpha )\geq \text{Nc}ʻ(\gamma \,\,\text{exp}\,\,\beta ) &\qquad \text{(1)}\\ \vdash .(1).*116·25.\supset \vdash . \text{Prop} \end{array} \]

[Pg 182]

The hypothesis is essential in the above proposition, for \(0^{0} = 1\) while \(0^{1} = 0\), so that \(0^{0} > 0^{1}\).

*117·591. \(\vdash :\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\varpi ^{\mu } \geq \varpi ^{\nu } \quad[*117·59]\)

*117·592. \(\vdash :\alpha ^{\delta } = 1.\alpha \neq 0.\alpha \neq 1.\supset .\delta = 0\)

Dem. \[ \begin{array}{l} \vdash .*116·203.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\delta \in \text{N}_{0}\text{C}:\\ [*117·551·53] &\supset :\alpha \geq 2:\delta \neq 0.\supset .\delta \geq 1:\\ [*117·581·591] &\supset :\delta \neq O.\supset .\alpha ^{\delta } \geq 2^{1}.\\ [*116·321.*117·244] &\supset .\alpha ^{\delta } \geq 2.\\ [*117·551] &\supset .\alpha ^{\delta } \neq 1 &\qquad \text{(1)}\\ \vdash .(1).\text{Transp}.\supset \vdash .\text{Prop} \end{array} \]

The above proposition is used in *120·53.

*117·6. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\mu +_{c}\nu \geq \mu .\mu +_{c}\nu \geq \nu\)

Dem. \[ \begin{array}{l} \vdash .*117·561·5.\supset \vdash :\text{Hp}.\supset .\mu +_{c}\nu \geq \mu +_{c}0.\mu +_{c}\nu \geq 0+_{c}\nu &\qquad \text{(1)}\\ \vdash .(1).*110·6.*117·244.\supset \vdash .\text{Prop} \end{array} \]

*117·61. \(\vdash :\nu > \mu .\supset .\mu +_{c} \nu > \mu \quad[*117·6·45]\)

*117·62. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\mu \times _{c}\nu \geq \mu .\mu \times _{c}\nu \geq \nu\)

Dem. \[ \begin{array}{l} \vdash .*117·571·53.\supset \vdash :\text{Hp}.\supset .\mu \times _{c}\nu \geq \mu \times _{c}1.\mu \times _{c}\nu \geq 1\times _{c}\nu &\qquad \text{(1)}\\ \vdash .(1).*113·621.*117·244.\supset \vdash .\text{Prop} \end{array} \]

*117·63. \(\vdash :\alpha ,\beta {\sim}\in 0\cup 1.\supset .\text{Nc}ʻ\alpha \times _{c}\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*52·4.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .(\exists x,x',y,y').x,x'\in \alpha .y,y'\in \beta .x \neq x'.y \neq y' &\qquad \text{(1)}\\ \vdash .*113·101. &\supset \vdash :\text{Hp}.x,x'\in \alpha .y,y'\in \beta .x \neq x'.y \neq y'.\rho = \downarrow yʻʻ\alpha .\\ &\sigma = x\downarrow ʻʻ(\beta -\iota ʻy)\cup \iota ʻx'\downarrow y'.\supset .\rho \cup \sigma \subset \beta \times \alpha &\qquad \text{(2)}\\ \vdash .*55·15.&\supset \vdash \colon\ldotp \text{Hp}(2).\supset :R\in \rho .\supset _{R}.\text{ᗡ}ʻR = \iota ʻy:\\ &S\in x\downarrow ʻʻ(\beta -\iota ʻy).\supset _{S}.\text{ᗡ}ʻS\in \beta -\iota ʻy:\text{ᗡ}ʻx'\downarrow y' = \iota ʻy':\\ [*51·23] &\supset :R\in \rho .S\in \sigma .\supset _{R,S}.\text{ᗡ}ʻR \neq \text{ᗡ}ʻS:\\ [*24·37.*30·37] &\supset :\rho \cap \sigma = \Lambda &\qquad \text{(3)}\\ \vdash .*73·61·611. &\supset \vdash :\text{Hp}(2).\supset .\rho \text{ sm } \alpha .x\downarrow ʻʻ(\beta -\iota ʻy) \text{ sm } (\beta -\iota ʻy) &\qquad \text{(4)}\\ \vdash .*55·202. &\supset \vdash :\text{Hp}(2).\supset .x'\downarrow y'{\sim}\in x\downarrow ʻʻ(\beta -\iota ʻy) &\qquad \text{(5)}\\ \vdash .(4).(5).*73·71. &\supset \vdash :\text{Hp}(2).\supset .\rho \text{ sm } \alpha .\sigma \text{ sm } \beta &\qquad \text{(6)}\\ \vdash .(3).(6).*110·13. &\supset \vdash :\text{Hp}(2).\supset .\rho \cup \sigma \in \text{Nc}ʻ(\alpha +\beta ) &\qquad \text{(7)}\\ \vdash .(2).(7).*117·221. &\supset \vdash :\text{Hp}(2).\supset .\text{Nc}ʻ(\beta \times \alpha ) \geq \text{Nc}ʻ(\alpha +\beta ) &\qquad \text{(8)}\\ \vdash .(1).(8).*113·141·25.*110·3.\supset \vdash .\text{Prop} \end{array} \]

[Pg 183]

*117·631. \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}-\iota ʻ0-\iota ʻ1.\supset .\mu \times _{c}\nu \geq \mu +_{c}\nu \quad[*117·63]\)

The two following propositions are lemmas for *117·64.

*117·632. \[\begin{align}&\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\kappa {\sim}\in 0\cup 1.\rho ,\sigma \in \text{Prod}ʻ\kappa .\rho \cap \sigma =\Lambda .\\ &T=\hat{\mu} \hat{x} \{(\exists \alpha ,\beta ).\alpha ,\beta \in \kappa .\alpha \neq \beta .x\in \beta .\mu =(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\cup \iota ʻx\}.\\ &\supset .T\in 1\rightarrow 1.\text{D}ʻT\subset \text{Prod}ʻ\kappa .\text{ᗡ}ʻT=sʻ\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*115·11·145. &\supset \vdash \colon\ldotp \text{Hp}.\alpha ,\beta \in \kappa .\alpha \neq \beta .\supset :\rho -\alpha -\beta \in \text{Prod}ʻ(\kappa -\iota ʻ\alpha -\iota ʻ\beta ):\\ [*115·11·145] &\supset :(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\in \text{Prod}ʻ(\kappa -\iota ʻ\beta ):\\ [*115·145] &\supset :x\in \beta .\supset .(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\cup \iota ʻx\in \text{Prod}ʻ\kappa &\qquad \text{(1)}\\ \vdash .(1).*21·33. &\supset \vdash :\text{Hp}.\mu Tx.\supset .\mu \in \text{Prod}ʻ\kappa &\qquad \text{(2)}\\ \vdash .*52·4.\text{Transp}. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \kappa .x\in \beta .\supset .(\exists \alpha ).\alpha \in \kappa .\alpha \neq \beta .\\ [*21·33.*33·131] &\supset .x\in \text{ᗡ}ʻT &\qquad \text{(3)}\\ \vdash .*21·33.*33·131. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x \in\text{ᗡ}ʻT.\supset .(\exists \beta ).\beta \in \kappa .x\in \beta &\qquad \text{(4)}\\ \vdash .(3).(4). &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻT=sʻ\kappa &\qquad \text{(5)}\\ \vdash :*21·33.*13·172. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu Tx.\nu Tx.\supset .\mu =\nu &\qquad \text{(6)}\\ \vdash .*21·33.*13·171. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu Tx.\mu Tx'.\supset .\\ &(\exists \alpha ,\alpha',\beta ,\beta').\alpha ,\alpha'\in \kappa .\beta ,\beta'\in \kappa .\alpha \neq \beta .\alpha'\neq \beta'.\\ &(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\cup \iota ʻx=(\rho -\alpha'-\beta')\cap (\sigma \cap \alpha')\cup \iota ʻx'.\\ [*24·48.\text{Hp}] &\supset .\iota ʻx=\iota ʻx' &\qquad \text{(7)}\\ \vdash .(2).(5).(6).(7).\supset \vdash . \text{Prop} \end{array} \]

*117·633. \[\begin{align}&\vdash \colon\ldotp \kappa \in \text{Cls}^{2} \text{excl}.\kappa {\sim}\in 0\cup 1:(\exists \rho ,\sigma ).\rho ,\sigma \in \text{Prod}ʻ\kappa .\rho \cap \sigma =\Lambda :\supset .\\ &\Pi \text{Nc}ʻ\kappa \geq \Sigma \text{Nc}ʻ\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*117·632.\supset \vdash :\text{Hp}.&\supset .(\exists \gamma ).\gamma \subset \text{Prod}ʻ\kappa .\gamma \text{ sm } sʻ\kappa .\\ [*117·221] &\supset .\text{Nc}ʻ\text{Prod}ʻ\kappa \geq \text{Nc}ʻsʻ\kappa &\qquad \text{(1)}\\ \vdash .(1).*115·12.*112·15.\supset \vdash . \text{Prop} \end{array} \]

*117·64. \[\begin{align}\vdash \colon\ldotp \kappa \in \text{Cls}^{2} \text{excl}:(\exists \rho ,\sigma ).\rho ,\sigma \in \text{Prod}ʻ\kappa .\rho \cap \sigma =&\Lambda :\supset .\\ &\Pi \text{Nc}ʻ\kappa \geq \Sigma \text{Nc}ʻ\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*112·321.*114·21. \supset \vdash :\kappa \in 1.&\supset .\Pi \text{Nc}ʻ\kappa =\Sigma \text{Nc}ʻ\kappa &\qquad \text{(1)}\\ \vdash .*114·2.*112·3. \supset \vdash :\kappa \in 0.&\supset .\Pi \text{Nc}ʻ\kappa =1.\Sigma \text{Nc}ʻ\kappa =0.\\ [*117·51] &\supset .\Pi \text{Nc}ʻ\kappa >\Sigma \text{Nc}ʻ\kappa &\qquad \text{(2)}\\ \vdash .(1).(2).*117·633.\supset \vdash . \text{Prop} \end{array} \]

*117·651. \(\vdash :\alpha {\sim}\in 0\cup 1.\supset .(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta }\geq \text{Nc}ʻ\alpha x_{c}\text{Nc}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*52·4.\text{Transp}. &\supset \vdash :\text{Hp}.\supset .(\exists x,y).x,y\in \alpha .x\neq y &\qquad \text{(1)}\\ \vdash .*116·152.*55·23·202. \supset \vdash :x,y\in \alpha .x\neq y.\supset .&x\downarrowʻʻ\beta ,y\downarrowʻʻ\beta \in (\alpha \,\,\text{exp}\,\,\beta ).\\ &x\downarrowʻʻ\beta \cap y\downarrowʻʻ\beta =\Lambda &\qquad \text{(2)}\\ \vdash .*113·111. &\supset \vdash .\alpha \downarrow_{,,}ʻʻ\beta \in \text{Cls}^{2} \text{excl} &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*117·64.*113·1·141·25.*116·25.(*116·01).\supset \vdash . \text{Prop} \end{array} \]

[Pg 184]

*117·652. \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0-\iota ʻ1.\nu \in \text{N}_{0}\text{C}.\supset .\mu ^\nu \geq \mu \times _{c} \nu \quad[*117·651]\)

*117·66. \(\vdash .\text{Nc}ʻ\text{Cl}ʻ\alpha >\text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*102·72. &\supset \vdash .{\sim}(\exists \beta ).\beta \subset \alpha .\beta \text{ sm }\text{Cl}ʻ\alpha &\qquad \text{(1)}\\ \vdash .*100·6.*60·61. &\supset \vdash .\iotaʻʻ\alpha \subset \text{Cl}ʻ\alpha .\iotaʻʻ\alpha \text{ sm }\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*117·13. &\supset \vdash .\text{Prop} \end{array} \]

*117·661. \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .2^\mu >\mu \quad[*117·66.*116·72]\)

The above proposition is important.

*117·67. \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\exists !\text{Prod}ʻ\kappa .\supset .\text{Nc}ʻsʻ\kappa \geq \text{Nc}ʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*115·16·11.&\supset \vdash :\kappa \in \text{Cls}^{2} \text{excl}.\mu \in \text{Prod}ʻ\kappa .\supset .\mu \text{ sm }\kappa .\mu \subset sʻ\kappa \\ . [*117·22] &\supset .\text{Nc}ʻsʻ\kappa \geq \text{Nc}ʻ\kappa :\supset \vdash .\text{Prop} \end{array} \]

*117·68. \[\begin{align}\vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .T=\hat{P} &\hat{\rho} \{\rho \in \kappa .P=R\upharpoonright -\iota ʻ\rho \unicode{x228d} S\upharpoonright \iota ʻ\rho\}.\\ &\supset .T\in 1\rightarrow 1.\text{D}ʻT\subset {\in}_{\Delta}ʻ\kappa .\text{ᗡ}ʻT=\kappa\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*21·33.*13·172. &\supset \vdash \colon\ldotp \text{Hp}.\supset :PT\rho .QT\rho .\supset .P=Q &\qquad \text{(1)}\\ \vdash .*23·631. &\supset \vdash :\text{Hp}.\rho \in \kappa .\supset .(Tʻ\rho )\dot{\cap} S=S\upharpoonright \iota ʻ\rho :\\ [*13·17] &\supset \vdash :\text{Hp}.\rho ,\sigma \in \kappa .Tʻ\rho =Tʻ\sigma .\supset .S\upharpoonright \iota ʻ\rho =S\upharpoonright \iota ʻ\sigma .\\ [*35·65] &\supset .\iota ʻ\rho =\iota ʻ\sigma .\\ [*51·23] &\supset .\rho =\sigma &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .T\in 1\rightarrow 1 &\qquad \text{(3)}\\ \vdash .*21·33.*33·131. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻT=\kappa &\qquad \text{(4)}\\ \vdash .*80·36. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻT\subset {\in}_{\Delta}ʻ\kappa &\qquad \text{(5)}\\ \vdash .(3).(4).(5). \supset \vdash .\text{Prop} \end{array} \]

*117·681. \(\vdash :(\exists R,S).R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \geq \text{Nc}ʻ\kappa \quad [*117·68·22]\)

*117·682. \(\vdash :\kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa\)

Dem. \[ \begin{array}{l} \vdash .*80·65. &\supset \vdash \colon\ldotp \text{Hp}.\supset :R \in {\in}_{\Delta}ʻ\kappa .S\in {\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .R\unicode{x228d} S\in {\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ \vdash .*80·14. &\supset \vdash :R\in {\in}_{\Delta}ʻ\lambda .S\in {\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda .\\ [*33·33] &\supset .R\dot{\cap} S=\dot{\Lambda} .\\ [*25·4] &\supset .(R\unicode{x228d} S)\dot{-} S=R &\qquad \text{(2)}\\ \vdash .(2).*13·171. &\supset \vdash :Q,R\in {\in}_{\Delta}ʻ\lambda .S\in {\in}_{\Delta}ʻ(\lambda -\kappa ).Q\unicode{x228d} S=R\unicode{x228d} S.\supset .Q=R &\qquad \text{(3)}\\ \vdash .(1).(3). &\supset \vdash :\text{Hp}.S\in {\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .(\unicode{x228d} S)\upharpoonright {\in}_{\Delta}ʻ\kappa \in 1\rightarrow 1.(\unicode{x228d} S)ʻʻ{\in}_{\Delta}ʻ\kappa \subset {\in}_{\Delta}ʻ\lambda .\\ [*117·22] &\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa :\supset \vdash .\text{Prop} \end{array} \]

*117·683. \[\begin{align}\vdash \colon\ldotp \kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ(\lambda -\kappa ):(\exists R,S).&R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} :\supset .\\ &\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ\kappa \quad[*117·681·682]\end{align}\]

*117·684. \[\begin{align}\vdash :\kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ\lambda :(\exists R,S).&R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} :\supset .\\ &\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ\kappa \quad[*117·683.*88·22]\end{align}\]

The above proposition is used in *120·765.

The correlators established at various stages throughout Section B present certain analogies to each other, and they or others closely resembling them will be found to be the correlators required in relation-arithmetic (Part IV). We shall therefore here collect together the most important propositions hitherto proved on correlators.

When we have to deal with correlators of two different functions of a single class, as e.g. \({\in}_{\Delta}ʻ\kappa\) and \(\text{Prod}ʻ\kappa\), the correlator is usually \(\text{D}\) or \(\dot{s}\) or \(\dot{s} \mid \text{D}\), with a suitable limitation on the converse domain. Sometimes it is \(\breve{\iota} \mid \text{D}\) or \(\breve{\in}\mid \text{D}\). Thus for example the class \({\in}\unicode{x21A7}ʻʻ\kappa\), by means of which \(\Sigma ʻ\kappa\) is defined (*112), has double similarity with \(\kappa\) if \(\kappa\in \text{Cls}^{2} \text{excl}\) (*112·14); in this case, the double correlator is \(\breve{\iota} \mid \text{D}\) with its converse domain limited, i.e. \[ \vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\breve{\iota} \mid \text{D}\upharpoonright \Sigma ʻ\kappa \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\kappa ) . \] In the case of \(\text{Prod}ʻ\kappa\) and \({\in}_{\Delta}ʻ\kappa\), the correlator is \(\text{D}\), i.e. \[ \vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\text{D}\upharpoonright {\in}_{\Delta}ʻ\kappa \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\kappa ) . \] In the case of \({\in}_{\Delta}ʻsʻ\kappa\) and \({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\), the correlator is \(\dot{s}\mid \text{D}\), i.e. \[ \vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\dot{s} \mid \text{D}\upharpoonright {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \in ({\in}_{\Delta}ʻsʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa ). \] \(\dot{s} \mid \text{D}\) also correlates \({\in}_{\Delta}ʻ\kappa\) with \({\in}_{\Delta}ʻ\in \unicode{x21A7}ʻʻ\kappa\) (*85·61) and \(P_{\Delta }ʻ\alpha\) with \({\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\) (*85·53), and \(P_{\Delta }ʻsʻ\kappa\) with \({\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\) (*85·27·42) if \(\kappa \in \text{Cls}^{2} \text{excl}\).

The correlator of \((\alpha \uparrow \beta )_{\Delta }ʻ\beta\) with \((\alpha \,\,\text{exp}\,\,\beta)\) is \(\dot{s}\) (*116·131).

Another kind of correlators arises where we are given a correlator of \(\kappa\) and \(\lambda\), and we wish to construct a correlator for some associated classes \(Wʻ\kappa\) and \(Wʻ\lambda\) where we are given correlators of \(\alpha\) with \(\gamma\) and of \(\beta\) with \(\delta\), and we wish to construct a correlator of \(\alpha\unicode{x2640}\beta\) with \(\gamma \unicode{x2640}\delta\), where \(\unicode{x2640}\) is some double descriptive function in the sense of *38. In this case, the correlator will usually be of the form \(R\parallel \breve{S}\) (with a limited converse domain). Sometimes \(R\) and \(S\) will be identical; sometimes \(S\) will be \(R_{\in}\). Such correlators always depend upon

*55·61. \(\vdash :\text{E}!Rʻx.\text{E}!Sʻy.\supset .(R\parallel \breve{S} )ʻ(x\downarrow y)=(Rʻx)\downarrow (Sʻy)\)

together with the propositions *74·77 seq. giving cases in which \((R\parallel \breve{S})\upharpoonright \lambda\) is a one-one relation. It follows from *55·61 that if \(R\) and \(S\) are correlators whose converse domains include the domain and converse domain respectively[Pg 186] of a relation \(P\), then \((R\parallel\breve{S})ʻP\) will be a relation holding between \(Rʻx\) and \(Sʻy\) whenever \(P\) holds between \(x\) and \(y\). Examples of such correlators as \(R\parallel \breve{S}\) are

*112·153. \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright sʻ\in \unicode{x21A7}ʻʻ\lambda \in (\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\)

*113·127. \[\begin{align}\vdash :R\upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma .S\upharpoonright \delta \in &\beta \,\overline{\text{ sm }}\, \delta .\supset .\\ &(R\parallel \breve{S} )\upharpoonright (\delta \times \gamma )\in (\alpha \downarrow_{,,}ʻʻ\beta )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,}ʻʻ\delta )\end{align}\]

*113·65. \(\vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta =(\downarrow z\parallel \text{Cnv}ʻ\downarrow z)ʻʻ(\alpha \times \beta )\)

*114·51. \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\)

*116·192. \[\begin{align}\vdash :R\upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma .S\upharpoonright \delta \in &\beta \,\overline{\text{ sm }}\, \delta .\supset .\\ &(R\parallel \breve{S} )\upharpoonright (\delta \times \gamma )\in (\alpha \,\,\text{exp}\,\,\beta )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\,\delta ).\\ &(R\parallel \breve{S} )_{\in }\upharpoonright (\gamma \,\,\text{exp}\,\,\delta )\in (\alpha \,\,\text{exp}\,\,\beta )\,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\,\delta )\end{align}\]

An exceptionally simple correlator is given by

*115·502. \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\)

Another exceptionally simple case is

*73·63. \[\begin{align}\vdash :S\in \alpha \,\overline{\text{ sm }}\, \beta .T\upharpoonright \alpha ,T\upharpoonright \beta \in 1\rightarrow 1.\alpha \cup &\beta \subset \text{ᗡ}ʻT.\supset .\\ &T\mid S\mid \breve{T} \in (Tʻʻ\alpha )\,\overline{\text{ sm }}\, (Tʻʻ\beta )\end{align}\]

By means of the above correlators, most correlators that are required can be calculated. Thus it will be seen that *116·192 in the above list is an immediate consequence of *113·127 and *115·502, since \[ \alpha \,\,\text{exp}\,\,\beta =\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta\,\text{and}\,sʻ\text{Prod}ʻ\gamma \downarrow_{,,}ʻʻ\delta =\delta \times \gamma . \]

In order to develop the subject, it is almost always necessary, not merely to prove that two classes are similar, but actually to construct a correlator of the two classes. This applies equally to relation-arithmetic, in which analogous correlators are used to prove ordinal similarity.

Summary of Section C.

The distinction of finite and infinite is not required, as appears from Section B, for the definition of the arithmetical operations or for the proof of their formal laws. There are, however, many important respects in which finite cardinals and classes differ respectively from infinite cardinals and classes, and these differences must now be investigated.

There are two different ways in which we may define the finite and the infinite, and these two ways cannot (so far as is known at present) be shown to be equivalent except by assuming the multiplicative axiom. As there seems no good reason for regarding one of these ways as giving more exactly than the other what is usually meant by the words "finite" and "infinite," we shall, to avoid confusion, give other names than these to each of the two ways of dividing classes and cardinals. The division effected by the first method of definition we shall call the division into inductive and non-inductive; that effected by the second method we shall call the division into non-reflexive and reflexive.

The division into inductive and non-inductive, which is treated in *120, is defined as follows. An inductive cardinal is one which can be reached from 0 by successive additions of 1; that is, an inductive cardinal is one which has to 0 the relation \((+_{c}1)_{*}\), where (by *38·02) \(+_{c}1\) is the relation of \(\alpha +_{c}1\) to \(\alpha\), and the subscript asterisk has the meaning defined in *90. Hence we put \[ \text{NC induct}=\hat{\alpha}\{\alpha (+_{c}1)_{*}0\} \quad\text{Df}. \] By applying the definition of *90, this gives \[ \vdash \colon\colon \alpha _{\in }\text{NC induct}.\equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.\xi +_{c}1\in \mu :0\in \mu :\supset _{\mu }.\alpha \in \mu . \] This proposition may be regarded as stating that an inductive cardinal is one which obeys mathematical induction starting from 0, i.e. it is one which possesses every property possessed by 0 and by the numbers obtained by adding 1 to numbers possessing the property. In elementary mathematics, it is customary to regard mathematical induction, as applied to the series of natural numbers, as a principle rather than a definition, but according to[Pg 188] the above procedure it becomes a definition rather than a principle. This procedure is unavoidable as soon as it is perceived that there are cardinals which do not obey mathematical induction starting from 0. (This only holds on the assumption that the total number of objects in any one type is not one of the inductive cardinals. This assumption, in a slightly different form, is introduced below as the "axiom of infinity.") Thus for example \(0\neq 1\), and \(\xi \neq \xi +_{c}1.\supset .\xi+_{c}1\neq \xi +_{c}2\). Hence if \(\alpha\) is any inductive cardinal, \(\alpha \neq \alpha +_{c}1\). But we know that \(\aleph _{0}\), the first of Cantor's transfinite cardinals[6], satisfies \(\aleph_{0}=\aleph _{0}+_{c}1\). Thus mathematical induction starting from 0 cannot be validly applied to prove properties of \(\aleph _{0}\). It follows that the inductive cardinals as above defined are only some among cardinals; nor does it appear that there is any way of defining them except as those that obey mathematical induction starting from 0. It follows that mathematical induction is not a principle, to be either proved or assumed as an axiom, but is merely a characteristic defining a certain class of cardinals, namely the class of inductive cardinals.

By a syllogism in Barbara, it is evident that 0 is an inductive cardinal; hence by the definition 1 is an inductive cardinal, and hence 2, 3, ... are inductive cardinals. Thus any given cardinal in the series of natural numbers can be shown to be an inductive cardinal. The usual elementary properties of inductive cardinals, such as the uniqueness of subtraction and division, are easily proved by mathematical induction.

We define an inductive class as a class the number of whose terms is an inductive cardinal. More simply, we put \[ \text{Cls induct}=sʻ\text{NC induct} \quad\text{Df}. \] It is then easily shown that an inductive class is one which can be reached from \(\Lambda\) by successive additions of single members. That is, if we put \[ \begin{aligned} M=\hat{\eta} \hat{\zeta} \{(\exists y).\zeta =\eta \cup \iota ʻy\},\\ \text{then}\quad \text{Cls induct}=\overleftarrow{M}_{*}ʻ\Lambda . \end{aligned} \]

Thus we have \[ \vdash \colon\colon \rho \in \text{Cls induct}.\equiv \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset _{\mu }.\rho \in \mu . \] We might equally well have begun by defining inductive classes, and proceeded to define inductive cardinals as the cardinals of inductive classes; in that case, we should have used the above relation \(M\) to define inductive classes.

Some of the properties which we expect inductive cardinals to possess, such for example as \(\alpha \neq \alpha +_{c}1\), can only be proved by assuming that no inductive cardinal is null, i.e. that \[ \alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha . \] This amounts to the assumption that, in any fixed type, a class can be found[Pg 189] having any assigned inductive number of terms. If this were false, there would have to be some definite member of the series of natural numbers which gave the total number of objects of the type in question. Thus suppose there were exactly n individuals in the universe, and no more, where n is an inductive cardinal. We should then have \(2^{n}\) classes, \(2^{2^{n}}\) classes of classes, and so on. In that case, in the type of individuals we should have \(n+_{c}1=\Lambda\), \(n+_{c}2=\Lambda\), etc. Hence we should have \[ n+_{c}1=(n+_{c}1)+_{c}1,\,\text{etc}. \] In the type of classes, we should get similar results for \(2^{n}\), and so on. It is plain (though not demonstrable except in each particular case) that if the assumption \(\alpha \in \text{NC induct}.\supset_{\alpha }.\exists !\alpha\) fails in any one type, it fails in any other type in the same hierarchy, and if it holds in any one, it holds in any other; for if n be the total number of individuals, then if \(n\) is an inductive cardinal, the total number of any other type is an inductive cardinal, while if \(n\) is not an inductive cardinal, no more is the total number of any other type. Hence the assumption \(\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha\) is either true in any type or false in any type in one hierarchy. We shall call it the "axiom of infinity," putting \[ \text{Infin ax}.=:\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha \quad\text{Df}. \] This assumption, like the multiplicative axiom, will be adduced as a hypothesis whenever it is relevant. It seems plain that there is nothing in logic to necessitate its truth or falsehood, and that it can only be legitimately believed or disbelieved on empirical grounds. When we wish to use a typically definite form of the axiom, we shall employ the definition \[ \text{Infin ax}(x).=:\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha (x) \quad\text{Df} \] which asserts that, if \(\alpha\) is any inductive cardinal, there are at least \(\alpha\) terms of the same type as \(x\).

It is important to observe that, although the axiom of infinity cannot (so far as appears) be proved a priori, we can prove that any given inductive cardinal exists in a sufficiently high type. For if the total number of individuals be \(n\), the numbers of objects in succeeding types are \(2^{n}\), \(2^{2^{n}}\), etc., and these numbers grow beyond any assigned inductive cardinal. Owing, however, to the fact that we cannot add together an infinite number of classes whose types increase without limit, we cannot hence show that there is a type in which every inductive cardinal exists, though we can show of every inductive cardinal that there is a type in which it exists. i.e. if \(\alpha\) is any inductive cardinal, there must be a type for \(x\) such that \(\exists !\alpha (x)\) is true; but there need not be a type for \(x\) such that if \(\alpha\) is any inductive cardinal, \(\exists!\alpha (x)\) is true.

The axiom of infinity suffices to prove the existence, in appropriate types, of \(\aleph _{0}\), \(2^{\aleph _{0}}\), \(2^{2^{{}\aleph_{0}}}\), ... \(\aleph _{1}\), \(\aleph _{2}\), ...[7]. It does not [Pg 190]suffice, so far as we know, to prove the existence of \(\aleph_{\omega}\) or any Aleph with a greater suffix than \(\omega\), because the existences of \(\aleph _{1}\), \(\aleph _{2}\), ... are proved in successively rising types, and no meaning can be found for a type whose order is infinite.

The other definition of finite and infinite is of less importance in practice than the definition by induction. It is dealt with in *124. According to this definition, we call a class reflexive when it contains a proper part similar to itself, i.e. we put \[ \text{Cls refl}=\hat{\alpha} \{(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR\subset \alpha .\text{ᗡ}ʻR\neq \alpha\} \quad\text{Df}, \] or, what comes to the same thing, \[ \text{Cls refl}=\hat{\alpha}\{(\exists R).R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !\overrightarrow{B}ʻR.\alpha =\text{D}ʻR\} \quad\text{Df}. \] We call a cardinal reflexive when it is the homogeneous cardinal of a reflexive class, i.e. we put \[ \text{NC refl} = N_{0}ʻʻ\text{Cls refl} \quad\text{Df}. \] It is easy to show that \[ \text{NC refl} =\hat{\alpha} \{\exists !\alpha .\alpha =\alpha +_{c}1\}. \] We find that inductive classes and cardinals are non-reflexive, and reflexive classes and cardinals are non-inductive. We find also that reflexive cardinals are those that are equal to or greater than \(\aleph _{0}\), while inductive cardinals are those that are less than \(\aleph _{0}\). By assuming the multiplicative axiom, we can show that every cardinal is equal to, greater than, or less than \(\aleph_{0}\), whence it follows that every cardinal is either reflexive or inductive, thus identifying the two definitions of finite and infinite. But so long as we refrain from assuming either the multiplicative axiom or some special axiom ad hoc, it remains possible (so far as is known at present) that there may be cardinals neither greater than, nor equal to, nor less than \(\aleph _{0}\). Such cardinals, if they exist, are neither inductive nor reflexive: they are infinite if we define infinity by the negation of induction, but finite if we define infinity by reflexiveness. It is possible that further investigation may either prove or disprove the existence of such cardinals; for the present, their existence must remain an open question, except for those who regard the multiplicative axiom as a self-evident truth.

In *121 we shall consider intervals in a discrete series; i.e. in a series generated by a one-one relation between consecutive terms. If \(P\) be the generating relation of such a series, and \(x\) and \(y\) be two members of the series, of which \(y\) is the later, the terms which lie between \(x\) and \(y\) are the terms \(z\) for which we have \[ xP_{\text{po}}z.zP_{\text{po}}y, \] where \(P_{\text{po}}\) has the meaning defined in *91. Hence we put \[ P(x-y)=\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy \quad\text{Df}, \] where "\(P(x-y)\)" means "the \(P\)-interval between \(x\) and \(y\)." We want[Pg 191] also symbols for the interval together with one or both of its end-points. For these we put \[ \begin{aligned} P(x\unicode{x22A3} y)&=\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{*}ʻy &\quad\text{Df},\\ P(x\unicode{x27dd}y)&=\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy &\quad\text{Df},\\ P(x\vdash\dashv y)&=\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{*}ʻy &\quad\text{Df}. \end{aligned} \] [8] Thus, for example, if \(x\) and \(y\) be inductive cardinals, and \(P\) be the relation of \(n\) to \(n+_{c}1\), and \(x < y\), \(P(x-y)\) will be the numbers greater than \(x\) and less than \(y\), while \(P(x\unicode{x27dd} y)\) will be these numbers together with \(y\), \(P(x\unicode{x22A2}y)\) will be these numbers together with \(x\), and \(P(x\vdash\dashv y)\) will be these numbers together with both \(x\) and \(y\). By means of intervals, we define a class of relations \(P_{\nu}\) (where \(\nu\) is any inductive cardinal), where "\(xP_{\nu }z\)" means that we can pass from \(x\) to \(z\) in \(\nu\) steps. In order to fit the case in which \(x\) and \(z\) are identical, and to insure that no relation such as \(P_{\nu}\) shall hold between terms which do not both belong to the field of \(P\), we put \[ P_{\nu }=\hat{x} \hat{y} \{\text{Nc}ʻP(x\vdash\dashv y)=\nu +_{c}1\} \quad\text{Df}. \] Then, provided \(P_{\text{po}}\,\unicode{x2abd}\, J\), \(P_{0}=I\upharpoonright CʻP\), and if further \(P\in 1\rightarrow 1\), then \(P_{1}= P\), \(P_{2}=P^{2}\), etc. If \(P\) is a transitive serial relation, \(P_{1}\) is the relation "immediately preceding," which has great importance in well-ordered series. In this case, \(P_{1}=P\dot{-} P^{2}\). If \(P\) is a transitive serial relation generating a finite series or a progression or a series of the type of the negative and positive integers in order of magnitude, we have \[ P=(P_{1})_{\text{po}}. \]

In *121 we shall only consider \(P_{\nu}\) in the case where \[ P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1), \] and generally we shall have the further hypothesis \(P_{\text{po}}\,\unicode{x2abd}\, J\). We can then prove that the interval between \(x\) and \(y\) is always an inductive class (it will be null unless \(xP_{*}y\)); this proposition is useful in its application to the number-series and to progressions generally.

When \(P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J\), [Pg 192]the class of such relations as \(P_{\nu }\) (where \(\nu\) is an inductive cardinal) is identical with \(\text{Potid}ʻP\), the class of powers of \(P\) (cf. *91 seq.). This identification (which does not hold in general without the above hypothesis) leads to many useful propositions. In *91 seq., we treated powers of a relation without the use of numbers, i.e. without defining the \(\nu\)th power of \(P\). When the powers of \(P\) are the class of such relations as \(P_{\nu }\), we can of course take \(P_{\nu }\) as the \(\nu\)th power of \(P\). The general definition of the \(\nu\)th power of \(P\) (where \(\nu\) is an inductive cardinal) will be given later, in *301; we shall denote it by \(P^\nu\), thereby including the notation \(P^{2}\) already defined.

In *122 we shall deal with progressions, i.e. with series of the type of the series of natural numbers. In this number, we shall deal with such series as generated by one-one relations; they will be dealt with at a later stage (*263) as generated by transitive relations. We define a progression as a one-one relation whose domain is the posterity of its first term, i.e. \[ \text{Prog} = (1 \rightarrow 1) \cap \hat{R} (\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR) \quad\text{Df}. \] According to this definition, there must be a first term \(BʻR\); \(\text{ᗡ}ʻR\) will be \(\breve{R} ʻʻ\overleftarrow{R}_{*}ʻBʻR\), i.e. \(\overleftarrow{R}_{\text{po}}ʻBʻR\), which is contained in \(\overleftarrow{R}_{*}ʻBʻR\), i.e. in \(\text{D}ʻR\); since \(\text{ᗡ}ʻR \subset \text{D}ʻR\), every term of the field of \(R\) has a successor, so that there is no end to the series; since \(CʻR = \text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR\), every term of the series can be reached from the beginning by successive steps. These characteristics suffice to define progressions.

In *123 we proceed to the definition and discussion of \(\aleph _{0}\), the smallest of reflexive cardinals. This is the cardinal number of any class whose terms can be arranged in a progression; hence it is the class of domains of progressions, i.e. we may put \[ \aleph _{0} = \text{D}ʻʻ\text{Prog} \quad\text{Df}. \] With this definition, remembering that \(\Lambda\) is a cardinal, we can prove that \(\aleph _{0}\) is a cardinal; but to prove that \(\aleph _{0}\) is an existent cardinal, we need the axiom of infinity. The existence-theorem for \(\aleph _{0}\) is then derived from the inductive cardinals, which, if no one of them is null, form a progression when arranged in order of magnitude. It should be observed that this existence-theorem is for a higher type than that for which the axiom of infinity is assumed. In order to get an existence-theorem for the same type, we need the multiplicative axiom as well.

After a number on reflexive classes and cardinals (*124) and a number on the axiom of infinity (*125), the Section ends with a number (*126) on "typically indefinite inductive cardinals." The constant inductive cardinals are the typically ambiguous symbols 0, 1, 2, ...; thus we want to define the class of inductive cardinals in such a way that a variable member of the class shall be typically ambiguous. This is not possible without a sacrifice of rigour, but in *126 it is shown how to minimize the sacrifice of rigour, and how to obviate the resulting logical dangers. A variable whose values are typically ambiguous is said to be "typically indefinite."

A proof that all inductive cardinals exist has often been derived from *120·57 (below). But according to the doctrine of types, this proof is invalid, since "\(\mu +_{0} 1\)" in *120·57 is necessarily of higher type than "\(\mu\)."

Summary of *118.

A difficulty arises respecting substitution in arithmetic. For if \(\mu\) is a formal number and its occurrence in \(f\mu\) is arithmetical, then by \(\text{IIT}\) \(\mu\) is always to be taken in an existential type. Hence we can only substitute a real variable \(\xi\) for \(\mu\) under the hypothesis \(\exists !\xi\), and we can only substitute another formal number \(\sigma\) for \(\mu\) provided that the equation \(\mu = \sigma\), which justifies the substitution, is arithmetical, i.e. provided that in this equation the type of \(\mu\) is such that \(\exists !\mu\).

The result is that the application of *20·18 is apt to lead to fallacies owing to the different meanings which a formal number may possess in different occurrences. Hitherto we have considered each case in detail, e.g. note on *110·61, and proof of *110·56.

The condition for the safe application of *20·18 is given in *118·01, namely

*118·01. \(\vdash \colon\ldotp \exists !\mu .\mu =\sigma .\supset :f\mu .\equiv .f\sigma \quad[*20·18]\)

This question is more fully discussed in the prefatory statement of this volume. The first reference to *118·01 is in *120·222. Another way of evading the difficulty is to work with formal numbers which, together with all their components, are of the same type. This leads to the consideration of Uniform Formal Numbers, which with the exception of *118·01 occupies the rest of the number.

The dominant type of a formal number as used in any context is the type of the formal number itself in that context, and the subordinate types of the formal number are the dominant types of its component formal numbers.

When the dominant types of some of the formal numbers are not expressly indicated by an explicit notation (cf. *65), the rules according to which the dominant types thus left ambiguous are to be related, so far as they are related, including the rules governing the relation of subordinate types, if left ambiguous, to dominant types, are given by conventions \(\text{IT}\), \(\text{IIT}\), and \(\text{AT}\) of the prefatory statement in this volume.

We have now to consider an important special case which arises when types are explicitly indicated by the use of *65·01·03. A formal number,[Pg 194] whose subordinate types are the same as its dominant type, is called uniform; and if some of its subordinate types are the same as its dominant type, it is called partially uniform. A formal number can only be partially uniform, or at least so designated as to be necessarily partially uniform, when the dominant type and those subordinate types identical with it are expressly indicated by *65·01·03. For otherwise the conventions \(\text{IT}\), \(\text{IIT}\), and perhaps also \(\text{AT}\), apply; and these do not secure uniformity, and may perhaps in some contexts be inconsistent with it.

Common sense in its consideration of arithmetic habitually disregards the possibility of a formal number representing \(\Lambda\). In other words, it always applies conventions \(\text{IIT}\) and \(\text{AT}\). But also, owing to its disregard of types, it assumes that the formal numbers are all uniform. The assumption which is really essential to this common sense reasoning, so far as the form of its arithmetical conclusions are concerned, is the assumption that none of the numerical symbols represent \(\Lambda\). This assumption is secured here, when no types are expressly indicated, by \(\text{IIT}\) and \(\text{AT}\). We have now to consider the effect on arithmetical operations of the other assumption, that the formal numbers are uniform, or partially uniform. There is no difficulty arising from any change of convention for symbolism, since, as stated above, partial or complete uniformity is secured by express indication of type. Accordingly conventions \(\text{IT}\), \(\text{IIT}\) continue, as always, to apply when the types of formal numbers are left ambiguous.

Convention \(\text{AT}\) will not be applied either in *118 or *119 or *120: in *118 the fact is entirely unimportant since the dominant types of equational occurrences are always indicated, so that no case arises when it could apply.

Apart from its intrinsic interest and its bearing on substitution, the arithmetic of uniform formal numbers is necessary for *120, where the fundamental arithmetical properties of inductive numbers are investigated.

The propositions of this number are proved by the use of the results of *117. The basis of the reasoning is

*118·13. \(\vdash \colon\ldotp \mu \leq \nu .\supset :\exists !\text{ sm }_{\xi }ʻʻ\nu .\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu\)

In *118·2·3·4 the meaning of the symbolism for dominant types is stated, namely

*118·2. \(\vdash .(\mu +_{c}\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\eta \text{ sm }_{\xi }(\alpha +\beta )\}\)

*118·3. \(\vdash .(\mu \times _{c}\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\eta \text{ sm }_{\xi }(\alpha \times \beta )\}\)

*118·4. \(\vdash .(\mu ^\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\eta \text{ sm }_{\xi }(\alpha \,\,\text{exp}\,\,\beta )\}\)

The important propositions which are finally reached for addition are

*118·23. \(\vdash :\mu ,\nu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu)_{\xi }\)

[Pg 195]

*118·24. \(\vdash :\nu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

*118·241. \(\vdash :\mu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\nu )_{\xi }\)

*118·25. \(\vdash .(\mu +_{c}\nu +_{c}\varpi )_{\xi }=\{(\mu +_{c}\nu )_{\xi }+_{c}\varpi\}_{\xi }=\{\mu +_{c}(\nu +_{c}\varpi )_{\xi }\}_{\xi }\)

The important propositions for multiplication are

*118·33. \(\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

*118·34. \(\vdash :\nu \in \text{NC}.\mu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

*118·341. \(\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu \times _{c}\nu )_{\xi }\)

*118·35. \(\vdash :\varpi \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{(\mu \times _{c}\nu )_{\xi }\times _{c}\varpi\}_{\xi }\)

*118·351. \(\vdash :\mu \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{\mu \times _{c}(\nu \times _{c}\varpi )_{\xi }\}_{\xi}\)

The important propositions for exponentiation are

*118·43. \(\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^{\text{ sm }_{\xi }ʻʻ\nu }\}_{\xi}\)

*118·44. \(\vdash :\nu \in \text{NC}.\mu \neq 0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=(\mu ^{\text{ sm }_{\xi }ʻʻ\nu })_{\xi }\)

*118·441. \(\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^\nu \}_{\xi}\)

*118·45. \(\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi )_{\xi }}\}_{\xi}\)

*118·451. \(\vdash :\varpi \neq 0.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=[\{(\mu ^\nu )_{\xi }\}^{\varpi}]_{\xi }\)

*118·46. \(\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{\mu ^{(\nu +_{c}\varpi )_{\xi }}\}_{\xi }\)

*118·461. \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^{\varpi })_{\xi }\}_{\xi }\)

with two analogous propositions *118·462·463,

*118·47. \(\vdash :\varpi \neq 0.\supset .{(\mu \times _{c}\nu )^{\varpi }}_{\xi }=[\{(\mu \times _{c}\nu )_{\xi }\}^{\varpi }]_{\xi }\)

*118·471. \[\begin{align}\vdash \colon\ldotp \mu \neq 0.\nu \neq 0.\lor.\varpi =0.\lor.{\sim}&(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\\ &\{(\mu \times _{c}\nu )^{\varpi }\}_{\xi }=\{(\mu ^{\varpi })_{\xi }\times _{c}(\nu ^{\varpi })_{\xi }\}_{\xi }\end{align}\]

with two analogous propositions *118·472·473.

It is thus seen that, apart from some exceptional cases connected with 0 and 1, in all arithmetical operations uniform, or partially uniform, formal numbers can replace those constructed in obedience to convention \(\text{IIT}\).

*118·01. \(\vdash \colon\ldotp \exists !\mu .\mu =\sigma .\supset :f\mu .\equiv .f\sigma \quad[*20·18]\)

[Pg 196]

As far as the symbolism is concerned, this proposition with the omission of \(\exists !\mu\) from the hypothesis is a transcript of *20·18. But if \(\mu\) or \(\sigma\) (not excluding both) is a formal number, \(\exists !\mu\) is required in case the occurrence of \(\mu\) in \(f\mu\) is arithmetical. In fact this proposition embodies the three fundamental propositions of the Principle of Arithmetical Substitution arrived at in the Prefatory Explanations on Types. Its necessity arises from the convention \(\text{IIT}\) which is explained there.

*118·11. \(\vdash :\exists !\text{Nc}(\xi )ʻ\beta .\alpha \subset \beta .\supset .\exists !\text{Nc}(\xi )ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*100·31.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ \gamma \in \text{Nc}(\xi )ʻ\beta .&\supset .\gamma \text{ sm }_{\xi }\beta .\\ [*73·1] &\supset .(\exists R).R\in 1(\xi )\rightarrow 1.\gamma =\text{D}ʻR.\beta =\text{ᗡ}ʻR.\\ [*22·55] &\supset .(\exists R).R\in 1(\xi )\rightarrow 1.\alpha \subset \text{ᗡ}ʻRʻʻ\alpha =Rʻʻ\alpha .\\ [*73·12] &\supset .(\exists R).Rʻʻ\alpha \text{ sm }_{\xi }\alpha .\\ [*100·31] &\supset .\exists !\text{Nc}(\xi )ʻ\alpha :\supset \vdash .\text{Prop} \end{array} \]

*118·12. \[\begin{align}&\vdash \colon\ldotp \text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .\supset :\exists !\text{Nc}(\xi )ʻ\beta .\supset .\exists !\text{Nc}(\xi )ʻ\alpha \\ &[*117·32·107.*100·511]\end{align}\]

*118·13. \(\vdash \colon\ldotp \mu \leq \nu .\supset :\exists !\text{ sm }_{\xi }ʻʻ\nu .\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu \quad[*117·32]\)

*118·2. \[\begin{align}&\vdash .(\mu +_{c}\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\eta \text{ sm }_{\xi }(\alpha +\beta )\}\\ &[(*65·01·03).*110·2]\end{align}\]

*118·201. \[\begin{align}&\vdash :\exists !(\mu +_{c}\nu ).\supset .\text{ sm }_{\xi }ʻʻ(\mu +_{c}\nu )=(\mu +_{c}\nu )_{\xi }\\ &[\text{*110·44. Note Erratum in enunciation}]\end{align}\]

*118·21. \(\vdash :\exists !(\mu +_{c}\nu )_{\xi }.\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu .\exists !\text{ sm }_{\xi }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash .*110·4.*118·2.\supset \vdash :\text{Hp}.&\supset .\mu ,\nu \in \text{N}_{0}\text{C}.\\ [*117·6] &\supset .\mu +_{c}\nu \geq \mu .\mu +_{c}\nu \geq \nu .\\ [*118·13·201.(\text{IIT})] &\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu .\exists !\text{ sm }_{\xi }ʻʻ\nu :\supset \vdash .\text{Prop} \end{array} \]

Here the reference \(\text{IIT}\) is to the convention \(\text{IIT}\) explained in the prefatory statement.

*118·22. \[\begin{align}\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :&\exists !(\mu +_{c}\nu )_{\xi }.\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }.\equiv .\\ &\exists !(\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }.\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\nu )_{\xi }\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*118·21.\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !(\mu +_{c}\nu )_{\xi }.&\equiv .\exists !(\mu +_{c}\nu )_{\xi }.\exists !\text{ sm }_{\xi }ʻʻ\mu .\exists !\text{ sm }_{\xi }ʻʻ\nu .\\ [*110·25·4] &\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(1)}\\ \vdash .*118·21.*103·43.*110·4.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\exists !(\mu +_{c}\nu )_{\xi }.&\equiv .\exists !(\mu +_{c}\nu )_{\xi }.\exists !\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu .\exists !\text{ sm }_{\xi }ʻʻ\nu .\\ [*103·43.*110·25·4] &\equiv .\exists !(\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ \text{Similarly}\qquad \vdash \colon\ldotp \text{Hp}.&\supset :\exists !(\mu +_{c}\nu )_{\xi }.\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\nu )_{\xi } &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*118·23. \(\vdash :\mu ,\nu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*118·21.*110·4·25.&\supset \vdash :\exists !(\mu +_{c}\nu )_{\xi }.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(1)}\\ \vdash .*118·22.&\supset \vdash :\text{Hp}.{\sim}\exists !(\mu +_{c}\nu )_{\xi }.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash .\text{Prop} \end{array} \]

[Pg 197]

*118·24. \(\vdash : \nu \in \text{NC} . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

Dem. \[ \begin{array}{l} \vdash . *118·21 . *110·4·25 . *103·43 . \supset \\ &\vdash : \exists ! (\mu +_{c} \nu )_{\xi } . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(1)}\\ \vdash . *110·4 . &\supset \vdash : \mu {\sim} \in \text{NC} . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ \vdash . *118·22 . &\supset \vdash : \text{Hp} . \mu \in \text{NC} . {\sim} \exists ! (\mu +_{c} \nu )_{\xi } . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(3)}\\ \vdash . (1) . (2) . (3) . \supset \vdash . \text{Prop} \end{array} \]

*118·241. \(\vdash : \mu \in \text{NC} . \supset . (\mu +_{c} \nu )_{\xi } = (\text{ sm }_{\xi }ʻʻ\mu +_{c} \nu )_{\xi } \quad[*118·24 . *110·51]\)

*118·25. \(\vdash . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c} \varpi\}_{\xi } = \{\mu +_{c} (\nu +_{c} \varpi )_{\xi }\}_{\xi}\)

Dem. \[ \begin{array}{l} \vdash . *110·42 . *118·241·201 . (\text{IIT}) . \supset \\ &\vdash : \mu , \nu \in \text{N}_{0}\text{C} . \supset . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c}\varpi\}_{\xi } &\qquad \text{(1)}\\ \vdash . *110·4 . &\supset \vdash : {\sim} (\mu , \nu \in \text{N}_{0}\text{C}) . \supset . \mu +_{c} \nu = \Lambda . (\mu +_{c} \nu )_{\xi } = \Lambda .\\ [*110·4] &\supset . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c} \varpi\}_{\xi } &\qquad \text{(2)}\\ \vdash . (1) . (2) . &\supset \vdash . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c} \varpi\}_{\xi } &\qquad \text{(3)}\\ \text{Similarly}\qquad &\vdash . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{\mu +_{c} (\nu +_{c} \varpi)_{\xi }\}_{\xi } &\qquad \text{(4)}\\ \vdash . (3) . (4) . &\supset \vdash . \text{Prop} \end{array} \]

*118·3. \[\begin{align}&\vdash . (\mu \times _{c} \nu )_{\xi } = \hat{\eta} \{(\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta . \eta \text{ sm }_{\xi } (\alpha \times \beta )\}\\ &[(*65·01·03) . *113·2]\end{align}\]

*118·301. \(\vdash : \exists ! (\mu \times _{c} \nu ) . \supset . \text{ sm }_{\xi }ʻʻ(\mu \times _{c} \nu ) = (\mu \times _{c} \nu )_{\xi } \quad[\text{Proof as in *118·201}]\)

*118·31. \(\vdash : \exists ! (\mu \times _{c} \nu )_{\xi } . \nu \neq 0 . \supset . \exists ! \text{ sm }_{\xi }ʻʻ\mu \)

Dem. \[ \begin{array}{l} \vdash . *101·15·12. &\supset \vdash : \mu = 0 . \supset . \exists ! \text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(1)}\\ \vdash . *113·203 . *118·3 . &\supset \vdash : \text{Hp} . \mu \neq 0 . \supset . \mu , \nu \in \text{N}_{0}\text{C} - \iota ʻ0 .\\ [*117·62] &\supset . \mu \times _{c} \nu \geq \mu .\\ [*118·13·301 . (\text{IIT})] &\supset . \exists ! \text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ \vdash . (1) . (2) . \supset \vdash . \text{Prop} \end{array} \]

*118·311. \(\vdash : \exists ! (\mu \times _{c} \nu )_{\xi } . \mu \neq 0 . \supset . \exists ! \text{ sm }_{\xi }ʻʻ\nu \quad[*118·31 . *113·27]\)

*118·32. \(\vdash \colon\ldotp \nu \in \text{NC} . \mu \neq 0 . \supset : \exists ! (\mu \times _{c} \nu )_{\xi } . \equiv . \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

Dem. \[ \begin{array}{l} \vdash . *113·203 . &\supset \vdash : \exists ! (\mu \times _{c} \nu )_{\xi } . \supset . \mu \in \text{NC} &\qquad \text{(1)}\\ \vdash . *113·203 . &\supset \vdash : \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } . \supset . \mu \in \text{NC} &\qquad \text{(2)}\\ \vdash . *113·203 . *118·311 . \supset \\ \vdash \colon\ldotp \text{Hp} . &\supset : \exists ! (\mu \times _{c} \nu )_{\xi } . \supset . \exists ! \mu . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ [*103·43] &\supset . \exists ! \text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ [(1) . *113·26 . *103·43] &\supset . \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(3)}\\ \vdash . *113·203 . *103·43 . \supset \\ \vdash \colon\ldotp \text{Hp} . &\supset : \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } . \supset . \exists ! \text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ [(2) . *113·26 . *103·43] &\supset . \exists ! (\mu \times _{c} \nu )_{\xi } &\qquad \text{(4)}\\ \vdash . (3) . (4) . \supset \vdash . \text{Prop} \end{array} \]

[Pg 198]

*118·33. \[\begin{align}&\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\\ &[\text{Proof as in *118·23, using *118·31·311. *113·203·26}]\end{align}\]

*118·34. \(\vdash :\nu \in \text{NC}.\mu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*118·311.*113·203·26.*103·43.\supset \\ &\vdash :\exists !(\mu \times _{c}\nu )_{\xi }.\mu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }&\qquad \text{(1)}\\ \vdash .*118·32.&\supset \vdash :\text{Hp}.{\sim}\exists !(\mu \times _{c}\nu )_{\xi }.\supset .{\sim}\exists !(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }.\\ [*24·51] &\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*118·341. \(\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{sm }_{\xi }ʻʻ\mu \times _{c}\nu )_{\xi } \quad[*118·34.*113·27]\)

*118·35. \[\begin{align}&\vdash :\varpi \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{(\mu \times _{c}\nu )_{\xi }\times _{c}\varpi \}_{\xi }\\ &[\text{Proof similar to *118·25, using *118·341·301.*113·203·23}]\end{align}\]

*118·351. \(\vdash :\mu \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{\mu \times _{c}(\nu \times _{c}\varpi )_{\xi }\}_{\xi } \quad[*118·35.*113·27]\)

*118·352. \[\begin{align}&\vdash :\mu \neq 0.\varpi \neq 0.\supset .\{\mu \times _{c}(\nu \times _{c}\varpi )_{\xi }\}_{\xi }=\{(\mu \times _{c}\nu )_{\xi }\times _{c}\varpi \}_{\xi }\\ &[*118·35·351]\end{align}\]

*118·4. \[\begin{align}&\vdash .(\mu ^\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\eta \text{ sm }_{\xi }(\alpha \,\,\text{exp}\,\,\beta )\}\\ &[(*65·01·03).*116·2]\end{align}\]

*118·401. \(\vdash :\exists !\mu ^\nu .\supset .\text{ sm }_{\xi }ʻʻ\mu ^\nu =(\mu ^\nu )_{\xi } \quad[\text{Proof as in *118·201}]\)

*118·402. \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\mu \neq 0.\mu \neq 1.\supset :\exists !(\mu ^\nu )_{\xi }.\supset .\exists !(\mu \times _{c}\nu )_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*103·2. \supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\alpha {\sim}\in 0\cup 1:\\ [*117·651] &\supset :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\\ &(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta }\geq \text{N}_{0}\text{c}ʻ\alpha \times _{c}\text{N}_{0}\text{c}ʻ\beta :\\ [*118·13·301·401.(\text{IIT})]&\supset :\exists !(\mu ^\nu )_{\xi }.\supset .\exists !(\mu \times _{c}\nu )_{\xi }\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*118·41. \(\vdash :\exists !(\mu ^\nu )_{\xi }.\nu \neq 0.\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu\)

Dem. \[ \begin{array}{l} \vdash .*118·402·31. \supset \vdash :\text{Hp}.\mu \neq 1.\mu \neq 0.\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(1)}\\ \vdash .*101·12·15·241·28.\supset \vdash \colon\ldotp \mu =0.\lor.\mu =1:\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*118·411. \(\vdash :\exists !(\mu ^\nu )_{\xi }.\mu \neq 0.\mu \neq 1.\supset .\exists !\text{ sm }_{\xi }ʻʻ\nu \quad[*118·402·311]\)

*118·42. \[\begin{align}&\vdash \colon\ldotp \nu \in \text{NC}.\mu \neq 0.\mu \neq 1.\supset :\exists !(\mu ^\nu )_{\xi }.\equiv .\exists !(\mu ^{\text{ sm }_{\xi }ʻʻ\nu })_{\xi }\\ &[\text{Proof as in *118·32, using *116·203·26.*118·411}]\end{align}\]

*118·421. \[\begin{align}&\vdash \colon\ldotp \mu \in \text{NC}.\nu \neq 0.\supset :\exists !(\mu ^\nu )_{\xi }.\equiv .\exists !\{(\text{ sm }_{\xi }ʻʻ\mu )^\nu\}_{\xi }\\ &[\text{Proof as in *118·32, using *116·203·26.*118·41}]\end{align}\]

*118·43. \[\begin{align}&\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^{\text{ sm }_{\xi }ʻʻ\nu }\}_{\xi }\\ &[\text{Proof as in *118·23, using *118·41·411.*116·203·26}]\end{align}\]

[Pg 199]

*118·44. \[\begin{align}&\vdash :\nu \in \text{NC}.\mu \neq 0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=(\mu ^{\text{ sm }_{\xi }ʻʻ\nu })_{\xi }\\ &[\text{Proof as in *118·34, using *116·203·26.*118·411·42}]\end{align}\]

*118·441. \[\begin{align}&\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^\nu\}_{\xi }\\ &[\text{Proof as in *118·34, using *116·203·26.*118·41·421}]\end{align}\]

*118·45. \(\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi )_{\xi }}\}_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*113·23.*118·44·301.(\text{IIT}).\supset\\ \vdash :\text{Hp}.\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi )_{\xi }}\}_{\xi } &&\qquad \text{(1)}\\ \vdash .*113·203.\supset \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).&\supset .\nu \times _{c}\varpi =\Lambda .\\ [*116·203] &\supset . (\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi )_{\xi }}\}_{\xi } &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*118·451. \(\vdash :\varpi \neq 0.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi } =[\{(\mu ^\nu )_{\xi }\}^\varpi ]_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*116·63. \supset \vdash :\text{Hp}.\mu \in \text{NC}.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi } &=\{(\mu ^\nu )^\varpi\}_{\xi }\\ [*116·23.*118·441·401.(\text{IIT})] &=[\{(\mu ^\nu )_{\xi }\}^\varpi ]_{\xi } &\qquad \text{(1)}\\ \vdash .*116·204.\supset \vdash :\mu {\sim}\in \text{NC}.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi } &=[\{(\mu ^\nu )_{\xi }\}^\varpi ]_{\xi } &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*118·46. \[\begin{align}&\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{\mu ^{(\nu +_{c}\varpi )}_{\xi}\}_{\xi }\\ &[\text{Proof as in *118·45, using *118·44·201.*116·203.*110·4·42}]\end{align}\]

*118·461. \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*116·52.\supset \vdash :\mu \neq 0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=(\mu ^\nu \times _{c}\mu ^\varpi )_{\xi }\\ [*116·35·23.*118·33·401.(\text{IIT})] =\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(1)}\\ \vdash .*110·4.*113·203.*116·203.\supset \\ \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(2)}\\ \vdash .*116·311.*113·601.*110·62.\supset \\ \vdash :\nu ,\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\mu =0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(3)}\\ \vdash .*116·311·301.*110·6.*113·601.\supset \\ \vdash :\nu \in \text{N}_{0}\text{C}-\iota ʻ0.\varpi =0.\mu =0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(4)}\\ \text{Similarly}\qquad \vdash :\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\nu =0.\mu =0.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(5)}\\ \vdash .*116·301.*113·621.\supset \\ \vdash :\nu =0.\varpi =0.\mu =0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(6)}\\ \vdash .(1).(2).(3).(4).(5).(6).\supset \vdash .\text{Prop} \end{array} \]

*118·462. \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{\mu ^\nu \times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } \quad[\text{Proof as in *118·461, using *118·34}]\)

[Pg 200]

*118·463. \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}\mu ^\varpi\}_{\xi } \quad[\text{Proof as in *118·461, using *118·341}]\)

*118·47. \[\begin{align}&\vdash :\varpi \neq 0.\supset .\{(\mu _{c}\nu )^\varpi\}_{\xi }=[\{(\mu \times _{c}\nu )_{\xi }\}^\varpi ]_{\xi }\\ &[\text{Proof as in *118·45, using *118·441}]\end{align}\]

*118·471. \[\begin{align}\vdash \colon\ldotp \mu \neq 0.\nu \neq 0.\lor.\varpi =0.\lor.{\sim}&(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\\ &\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}(\nu ^\varpi )_{\xi }\}_{\xi }\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*116·55.\supset \vdash :\mu \neq 0.\nu \neq 0.\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{\mu ^\varpi \times _{c}\nu ^\varpi\}_{\xi }\\ [*116·35·23.*118·33·401.(\text{IIT})] =\{(\mu ^\varpi )_{\xi }\times _{c}(\nu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(1)}\\ \vdash .*110·4.*113·203.*116·203.\supset\\ \vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}(\nu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(2)}\\ \vdash .*116·301.*113·621.\supset \\ \vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\varpi =0.\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}(\nu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*118·472. \[\begin{align}&\vdash \colon\ldotp \mu \neq 0.\lor.\varpi =0.\lor.{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{\mu ^\varpi \times _{c}\nu ^\varpi )_{\xi }\}_{\xi }\\ &[\text{Proof as in *118·471, using *118·34}]\end{align}\]

*118·473. \[\begin{align}&\vdash \colon\ldotp \nu \neq 0.\lor.\varpi =0.\lor.{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}\nu ^\varpi\}_{\xi }\\ &[\text{Proof as in *118·471, using *118·341}]\end{align}\]

Summary of *119.

The treatment of subtraction follows the same general lines as that of addition, and is simplified by the results in *110. A difficulty arises from the fact that subtraction (in any ordinary sense of the term) is not always possible; and also from the fact that the result, when possible, is not always a cardinal number.

We put

*119·01. \(\gamma -_{c}\nu =\hat{\xi} \{\text{Nc}ʻ\xi +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu\} \quad\text{Df}\)

Thus when subtraction (in the ordinary sense of the term) is not possible, \[ \gamma -_{c}\nu =\Lambda . \]

The question of existential adjustment of types is dealt with by \(\text{IIT}\) of the prefatory statement combined with the following definitions:

*119·02. \(\text{Nc}ʻ\alpha -_{c}\nu =\text{N}_{0}\text{c}ʻ\alpha -_{c}\nu \quad\text{Df}\)

*119·03. \(\gamma -_{c}\text{Nc}ʻ\beta =\gamma -_{c}\text{N}_{0}\text{c}ʻ\beta \quad\text{Df}\)

We then proceed to deduce the elementary properties derivable from these definitions.

*119·11. \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma ,\nu \in \text{N}_{0}\text{C}\)

*119·12 \(\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\equiv .\alpha \text{ sm }\xi +\beta\)

*119·14. \(\vdash :\xi \in \gamma -_{c}\nu .\supset .\text{N}_{0}\text{c}ʻ\xi \subset \gamma -_{c}\nu\)

*119·25. \(\vdash :\gamma \geq \nu .\supset .\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma\)

*119·26. \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma \geq \nu\)

The next group of propositions is concerned with some simple results of subtraction.

*119·32. \(\vdash :(\gamma +_{c}\nu )-_{c}\nu \in \text{N}_{0}\text{C}.\supset .\text{ sm }ʻʻ\gamma =(\gamma +_{c}\nu )-_{c}\nu\)

*119·34. \(\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma\)

*119·35. \(\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .\alpha +_{c}\gamma =(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )\)

Associative laws are then considered.

*119·44. \(\vdash :\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi\)

*119·45. \(\vdash :(\mu +_{c}\nu )-_{c}\varpi \in \text{NC}.\exists !\{\mu +_{c}(\nu -_{c}\varpi )\}.\supset .\mu +_{c}(\nu -_{c}\varpi )=(\mu +_{c}\nu )-_{c}\varpi\)

The question of types is then dealt with:

[Pg 202]

*119·52. \(\vdash :\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta }\cap \text{D}ʻ\text{ sm }_{\delta ,\gamma }\)

A difficulty arises from the fact that if \(\tau _{1}\) and \(\tau_{2}\) are two complete types whose members are classes, we cannot prove that, either \(\tau _{1} = \text{ sm }ʻʻ\tau _{2}\) or \(\tau _{2}= \text{ sm }ʻʻ\tau _{1}\). We put

*119·54. \(\text{SM}(\delta ,\gamma ).=:tʻ\delta =\text{D}ʻ\text{ sm }_{\delta ,\gamma }.\lor.tʻ\gamma =\text{D}ʻ\text{ sm }_{\gamma ,\delta } \quad\text{Df}\)

Then we obtain

*119·541. \[\begin{align}\vdash :\text{SM}(\delta ,\gamma ).(\mu -_{c}\nu )_{\gamma }&\in \text{N}_{0}\text{C}.(\mu -_{c}\nu )_{\delta }\in \text{NC}.\supset .\\ &\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta }\end{align}\]

Finally we show that any existential adjustment of types will suffice for the components:

*119·61. \(\vdash :\mu \in \text{N}_{0}\text{C}.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .\mu -_{c}\nu =\text{ sm }_{\xi }ʻʻ\mu -_{c}\nu\)

*119·62. \(\vdash :\nu \in \text{N}_{0}\text{C}.\exists !\text{ sm }_{\xi }ʻʻ\nu .\supset .\mu -_{c}\nu =\mu -_{c}\text{ sm }_{\xi }ʻʻ\nu\)

Also *119·25·26 are now extended to

*119·64. \(\vdash \colon\ldotp \exists !\text{ sm }_{\xi }ʻʻ\mu .\supset :\mu \geq \nu .\equiv .\exists !(\mu -_{c}\nu )_{\xi }\)

The only applications of the propositions of this number are in connection with Inductive Cardinals (cf. *120).

*119·01. \(\gamma -_{c}\nu =\hat{\xi} \{\text{Nc}ʻ\xi +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu\} \quad\text{Df}\)

Here the suffix to the sign of subtraction is introduced to show that we are concerned with cardinal subtraction. It will be found that \(\gamma -_{c}\nu\) is not an \(\text{NC}\) except under hypotheses for \(\gamma\) and \(\nu\).

*119·02. \(\text{Nc}ʻ\alpha -_{c}\nu =\text{N}_{0}\text{c}ʻ\alpha -_{c}\nu \quad\text{Df}\)

*119·03. \(\gamma -_{c}\text{Nc}ʻ\beta =\gamma -_{c}\text{N}_{0}\text{c}ʻ\beta \quad\text{Df}\)

*119·04. \(\vdash .\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha -_{c}\text{N}_{0}\text{c}ʻ\beta \quad[*119·02·03]\)

Note that the occurrence of a formal number in the place of \(\gamma\) or \(\nu\) in \(\gamma -_{c}\nu\) is an arithmetic occurrence, and accordingly \(\text{IIT}\) applies to it.

*119·1. \(\vdash :\xi \in \gamma -_{c}\nu .\equiv .\text{Nc}ʻ\xi +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu \quad [(*119·01)]\)

*119·101. \(\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\nu .\equiv .\text{Nc}ʻ\xi +_{c}\nu =\text{N}_{0}\text{c}ʻ\alpha \quad[(*119·02).*103·13]\)

*119·102. \[\begin{align}&\vdash :\xi \in \gamma -_{c}\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta\\ &[(*119·03).*110·3]\end{align}\]

*119·103. \[\begin{align}&\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha\\ &[*119·04.*110·3.*103·13]\end{align}\]

*119·11. \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma ,\nu \in \text{N}_{0}\text{C} \quad[*110·4·42.*103·34]\)

*119·12. \(\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\equiv .\alpha \text{ sm }\xi +\beta\)

Dem. \[ \begin{array}{l} \vdash .*119·103.\supset \vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .&\equiv .\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha .\\ [*110·3] &\equiv .\text{Nc}ʻ(\xi +\beta )=\text{N}_{0}\text{c}ʻ\alpha .\\ [*100·35.*103·13] &\equiv .\alpha \text{ sm }\xi +\beta :\supset \vdash .\text{Prop} \end{array} \]

[Pg 203]

Thus \(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta\) is an \(\text{NC}\) when \(\hat{\xi} (\alpha \text{ sm }\xi +\beta )\) is an \(\text{NC}\).

*119·13. \(\vdash :\text{N}_{0}\text{c}ʻ\gamma \subset \text{Nc}ʻa-_{c}\text{Nc}ʻ\beta .\equiv .\alpha \text{ sm }(\gamma +\beta )\)

Dem. \[ \begin{array}{l} \vdash .*22·1.\supset \vdash \colon\ldotp \text{N}_{0}\text{c}ʻ\gamma \subset \text{Nc}ʻa-_{c}\text{Nc}ʻ\beta .&\equiv :\xi \in \text{N}_{0}\text{c}ʻ\gamma .\supset _{\xi }.\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta :\\ [*103·12.*119·12] &\supset :\alpha \text{ sm }(\gamma +\beta ) &\qquad \text{(1)}\\ \vdash .*110·15.*100·31.&\supset \vdash \colon\ldotp \alpha \text{ sm }(\gamma +\beta ).\supset :\xi \in \text{N}_{0}\text{c}ʻ\gamma .\supset .(\xi +\beta )\text{ sm }(\gamma +\beta ).\\ [*73·32] &\supset .\alpha \text{ sm }(\xi +\beta ).\\ [*119·12] &\supset .\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*119·14. \(\vdash :\xi \in \gamma -_{c}\nu .\supset .\text{N}_{0}\text{c}ʻ\xi \subset \gamma -_{c}\nu \quad[*119·1.*100·31·321]\)

*119·21. \(\vdash :\beta \subset \alpha .\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }\)

The notation is defined in *65·01.

Dem. \[ \begin{array}{l} \vdash .*24·411·21.\supset \vdash :\text{Hp}.&\supset .\alpha =\beta \cup (\alpha -\beta ).\beta \cap (\alpha -\beta )=\Lambda .\\ [*110·32] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ(\alpha -\beta ).\\ [*10·24] &\supset .(\exists \xi ).\xi \in tʻ\alpha .\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ\xi .\\ [*119·103] &\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }:\supset \vdash .\text{Prop} \end{array} \]

*119·22. \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }\)

Dem. \[ \begin{array}{l} \vdash .*117·221.&\supset \vdash :\text{Hp}.\supset .(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta .\\ [*119·21] &\supset .(\exists \rho ).\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\rho )_{\alpha }.\rho \text{ sm }\beta .\\ [*100·35.*119·04] &\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }:\supset \vdash .\text{Prop} \end{array} \]

*119·23. \(\vdash :\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta ).\supset .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha\)

Dem. \[ \begin{array}{l} \vdash .*119·103.\supset \vdash :\text{Hp}.&\supset .(\exists \xi ).\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ\xi .\\ [*110·71] &\supset .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha :\supset \vdash .\text{Prop} \end{array} \]

*119·24. \(\vdash :\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta ).\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta \quad[*119·23.*117·221]\)

*119·25. \(\vdash :\gamma \geq \nu .\supset .\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash .*117·24.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta .\\ [*117·107] &\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\\ [*119·22·04] &\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !(\text{N}_{0}\text{c}ʻ\alpha -_{c}\text{N}_{0}\text{c}ʻ\beta )_{\alpha }.\\ [(*63·02).*13·193] &\supset ·\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma :\supset \vdash .\text{Prop} \end{array} \]

*119·26. \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma \geq \nu\)

Dem. \[ \begin{array}{l} \vdash .*119·11.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !(\text{N}_{0}\text{c}ʻ\alpha -_{c}\text{N}_{0}\text{c}ʻ\beta ).\\ [*119·04·24] &\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\\ [*117·107.*13·193] &\supset .\gamma \geq \nu :\supset \vdash .\text{Prop} \end{array} \]

*119·27. \(\vdash :\gamma \geq \nu .\equiv .\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \quad[*119·25·26]\)

For the extension of this theorem cf. *119·64.

[Pg 204]

*119·31. \(\vdash :\gamma ,\nu \in \text{N}_{0}\text{C}.\supset .\text{ sm }ʻʻ\gamma \subset (\gamma +_{c}\nu )-_{c}\nu\)

Dem. \[ \begin{array}{l} \vdash .*119·1.(\text{IIT}).\supset \vdash :\xi \in (\gamma +_{c}\nu )-_{c}\nu .&\equiv .\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu .\exists !\gamma +_{c}\nu &\qquad \text{(1)}\\ \vdash .*100·51·521. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \text{ sm }ʻʻ\gamma .\supset .\text{Nc}ʻ\xi =\gamma .\\ [*103·22.*118·01] &\supset .\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu .\\ [*110·22·03.*103·13] &\supset .\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu .\exists !\gamma +_{c}\nu .\\ [(1)] &\supset .\xi \in (\gamma +_{c}\nu )-_{c}\nu :\supset \vdash .\text{Prop} \end{array} \]

The penultimate step in the proof employs the principle, explained in the prefatory statement, that, since in the previous line the equation \[ \text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu \] has its sides undetermined in type by the conventions \(\text{IT}\) and \(\text{IIT}\), any convenient type can be chosen for them. The type chosen in this line is such that \(\exists !\gamma +_{c}\nu\), and the references indicate the existence of at least one such type.

*119·32. \[\begin{align}&\vdash :(\gamma +_{c}\nu )-_{c}\nu \in \text{N}_{0}\text{C}.\supset .\text{ sm }ʻʻ\gamma =(\gamma +_{c}\nu )-_{c}\nu\\ &[*119·11·31.*103·22.*100·52·42]\end{align}\]

*119·33. \(\vdash :\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta \in \text{N}_{0}\text{C}.\supset .(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*119·13.&\supset \vdash :\text{N}_{0}\text{c}ʻ\gamma =\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\supset .\alpha \text{ sm }(\gamma +\beta ) &\qquad \text{(1)}\\ \vdash .*20·18.*118·01.\supset \vdash \colon\ldotp \text{Hp}(1).\supset : (\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\xi .&\equiv _{\xi }.\text{Nc}ʻ\gamma +_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\xi .\\ [*110·3.*100·35] &\equiv _{\xi }.\xi \text{ sm }(\gamma +\beta ).\\ [(1).*103·42] &\equiv _{\xi }.\text{N}_{0}\text{c}ʻ\xi =\text{Nc}ʻ\alpha &\qquad \text{(2)}\\ \vdash .*103·2·34.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !\text{Nc}ʻ\alpha .\supset .(\exists \xi ).\text{N}_{0}\text{c}ʻ\xi =\text{Nc}ʻ\alpha .\\ [(2).*10·1] &\supset .(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha &\qquad \text{(3)}\\ \vdash .*110·42.*103·34·2.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ &\exists !\{(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta \}.\supset .(\exists \xi ).\text{N}_{0}\text{c}ʻ\xi =(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta .\\ [(2).*10·1] &\supset .(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*119·34. \[\begin{align}&\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma\\ &[*119·11·33.*103·2.*100·51.*118·01]\end{align}\]

*119·35. \(\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .\alpha +_{c}\gamma =(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )\)

Dem. \[ \begin{array}{l} \vdash .*110·51·56.\supset \vdash :\text{Hp}.\supset .(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )&=\alpha +_{c}\{(\gamma -_{c}\nu )+_{c}\nu\}\\ [*119·34] &=\alpha +_{c}\text{ sm }ʻʻ\gamma \\ [*118·24.*119·11] &=\alpha +_{c}\gamma :\supset \vdash .\text{Prop} \end{array} \]

*119·41. \[\begin{align}\vdash \colon\ldotp \delta \in \text{Nc}ʻ\beta& -_{c}\text{Nc}ʻ\gamma .\supset :\\ &\xi \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .{(\alpha +\delta )+\gamma }\text{ sm }(\xi +\gamma )\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*119·12.*110·3. &\supset \vdash :\xi \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .(\alpha +\beta )\text{ sm }(\xi +\gamma ) &\qquad \text{(1)}\\ \vdash .*119·12. &\supset \vdash :\text{Hp}.\equiv .\beta \text{ sm }(\delta +\gamma ) &\qquad \text{(2)}\\ \vdash .(1).(2).*110·15·53.\supset \vdash . \text{Prop} \end{array} \]

[Pg 205]

*119·42. \[\begin{align}\vdash \colon\ldotp \text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma &\in \text{N}_{0}\text{C}.\eta \in \text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma ).\supset :\\ &\xi \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .(\eta +\gamma )\text{ sm }(\xi +\gamma )\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*118·01.*110·3.*103·2.*100·31.&\supset \vdash \colon\ldotp \text{N}_{0}\text{c}ʻ\delta =\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma .\supset:\\ &\eta \in \text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma ).\equiv .\eta \text{ sm }(a+\delta ) &\qquad \text{(1)}\\ \vdash .*119·41.(1).*103·12.*110·15.&\supset \vdash .\text{Prop} \end{array} \]

Note that if \(\gamma\) be an infinite class, it does not follow from \((\eta +\gamma)\text{ sm }(\xi +\gamma)\) that \(\eta \text{ sm }\xi\). This will be proved, however, when \(\gamma\) is an inductive class (cf. *120·41).

*119·43. \[\begin{align}\vdash :\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma &\in \text{N}_{0}\text{C}.\supset .\\ &\text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma )\subset (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*119·42.\supset \vdash \colon\ldotp \text{Hp}.&\eta \in \text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma ).\supset :\\ &\eta \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .(\eta +\gamma )\text{ sm }(\eta +\gamma ):\\ [*73·3] \supset :&\eta \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c} \text{Nc}ʻ\gamma &\qquad \text{(1)}\\ \vdash .(1).*22·1.\supset \vdash .\text{Prop} \end{array} \]

*119·44. \(\vdash :\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi\)

Dem. \[ \begin{array}{l} \vdash .*119·11·43.*103·2.&\supset \\ \vdash : \nu -_{c} \varpi \in \text{N}_{0}\text{C}.\mu \in \text{N}_{0}\text{C}.&\supset .\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi &\qquad \text{(1)}\\ \vdash . *110·4·42.*119·11.&\supset \\ \vdash :{\sim}\{\nu -_{c}\varpi \in \text{N}_{0}\text{C}.\mu \in \text{N}_{0}\text{C}\}.&\supset .\mu +_{c}(\nu -_{c}\varpi )=\Lambda .\\ [*24·12] &\supset .\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*119·45. \[\begin{align}&\vdash :(\mu +_{c} \nu )-_{c}\varpi \in \text{NC}.\exists !\{\mu +_{c}(\nu -_{c}\varpi )\}.\supset .\mu +_{c}(\nu -_{c}\varpi )=(\mu +_{c}\nu )-_{c}\varpi \\ &[*119·44.*100·33·321.*110·42]\end{align}\]

*119·51. \(\vdash :\text{ sm }_{\delta ,\gamma }ʻʻ(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\gamma }=(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\delta }\cap \text{D}ʻ\text{ sm }_{\delta ,\gamma }\)

Dem. \[ \begin{array}{l} \vdash .*119·12. \supset \vdash :\eta \in (\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\gamma }.\zeta \text{ sm }_{\delta ,\gamma }\eta .&\equiv .\alpha \text{ sm }\eta +\beta .\zeta \text{ sm }_{\delta ,\gamma }\eta .\\ [*110·15] &\equiv .\alpha \text{ sm }\zeta +\beta .\zeta \text{ sm }_{\delta ,\gamma }\eta .\\ [*119·12] &\equiv .\zeta \in (\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\delta }.\zeta \text{ sm }_{\delta ,\gamma }\eta :\\ [*37·1.*33·13]\supset \vdash .\text{ sm }_{\delta ,\gamma }ʻʻ(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\gamma }&=(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\delta }\cap \text{D}ʻ\text{ sm }_{\delta ,\gamma }:\supset \vdash .\text{Prop} \end{array} \]

*119·52. \(\vdash :\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta }\cap \text{D}ʻ\text{ sm }_{\delta ,\gamma } \quad[*119·51·11]\)

The difficulty in respect to types, which arises from the fact that \(\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }\) and \((\mu-_{c}\nu )_{\delta }\) have not been proved to be identical, does not exist when \(\nu\) is an "inductive number"; cf. *120·413.

*119·53. \(\vdash \colon\ldotp tʻ\delta =\text{D}ʻ\text{ sm }_{\delta ,\gamma }.\supset :\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta } \quad[*119·52.(*65·01)]\)

*119·531. \(\vdash :tʻ\delta =\text{D}ʻ\text{ sm }_{\delta ,\gamma }.(\mu -_{c}\nu )_{\delta }\in \text{N}_{0}\text{C}.\supset .\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta }\in \text{N}_{0}\text{C}\)

Dem. \[ \begin{array}{l} \vdash .*65·13. \supset \vdash :\text{Hp}.&\supset .(\mu -_{c}\nu )_{\delta }\subset \text{D}ʻ\text{ sm }_{\delta ,\gamma }.\\ [*37·43.*103·22.(*65·1)]&\supset .\exists !\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta }.\\ [*100·52.*103·34] &\supset .\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta }\in \text{N}_{0}\text{C}:\supset \vdash .\text{Prop} \end{array} \]

[Pg 206]

*119·532. \[\begin{align}\vdash : tʻ\delta = \text{D}ʻ\text{ sm }_{\delta ,\gamma }.(\mu -_{c}\nu )_{\delta } \in \text{N}_{0}\text{C}.&(\mu -_{c}\nu )_{\gamma } \in \text{NC}.\supset .\\ &\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta } = (\mu -_{c}\nu )_{\gamma }\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *119·52·531 .\supset \vdash : \text{Hp} . &\supset . \exists ! (\mu -_{c}\nu )_{\gamma }.\\ [*119·52·531·*100·34 &\supset . \text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta } = (\mu -_{c}\nu )_{\gamma } : \supset \vdash . \text{Prop} \end{array} \]

*119·54. \(\text{SM} (\delta ,\gamma ). = : tʻ\delta = \text{D}ʻ\text{ sm }_{\delta ,\gamma } . \lor . tʻ\gamma = \text{D}ʻ\text{ sm }_{\gamma ,\delta } \quad\text{Df}\)

*119·541. \[\begin{align}\vdash : SM(\delta ,\gamma ).(\mu -_{c}\nu )_{\gamma } \in &\text{N}_{0}\text{C} . (\mu -_{c}\nu )_{\delta } \in \text{NC}.\supset .\\ &\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma } = (\mu -_{c}\nu )_{\delta } \quad[*119·53·532]\end{align}\]

*119·61. \(\vdash : \mu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\mu . \supset . \mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c}\nu\)

Dem. \[ \begin{array}{l} \vdash . *119·1 .\supset \vdash \colon\ldotp \text{Hp}.\supset : \eta \in \mu -_{c}\nu .&\equiv .\text{Nc}ʻ\eta +_{c}\nu = \mu . \exists ! \mu .\\ [*103·16.*118·201.*37·29] &\equiv .(\text{Nc}ʻ\eta +_{c}\nu )_{\xi } = \text{ sm }_{\xi }ʻʻ\mu .\\ [*119·1] &\equiv . \eta \in \text{ sm }_{\xi }ʻʻ\mu -_{c}\nu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*119·62. \(\vdash : \nu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\nu . \supset .\mu -_{c}\nu = \mu -_{c} \text{ sm }_{\xi }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash . *119·1 .\supset \vdash \colon\ldotp \text{Hp}.\supset : \eta \in \mu -_{c}\nu .&\equiv .\text{Nc}ʻ\eta +_{c}\nu = \mu . \exists ! \mu .\\ [*110·25] &\equiv .\text{Nc}ʻ\eta +_{c} \text{ sm }_{\xi }ʻʻ\nu = \mu .\exists !\mu .\\ [*119·1] &\equiv . \eta \in \mu -_{c}\text{ sm }_{\xi }ʻʻ\nu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*119·63. \(\vdash : \mu , \nu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\mu .\supset .\mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c} \text{ sm }_{\xi }ʻʻ\nu\)

Dem. \[ \begin{array}{l} \vdash . *119·26 . &\supset \vdash : \text{Hp} . \exists !\mu -_{c}\nu . \supset . \mu \geq \nu .\\ [*118·13] & \supset . \exists !\text{ sm }_{\xi }ʻʻ\nu .\\ [*119·61·62] &\supset . \mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c}\text{ sm }_{\xi }ʻʻ\nu &\qquad \text{(1)}\\ \vdash . *119·11.*103·13. \supset \\ \vdash : \text{Hp}.\exists ! \text{ sm }_{\xi }ʻʻ\mu -_{c}\text{ sm }_{\xi }ʻʻ\nu . &\supset . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ [*119·61·62] &\supset . \mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c} \text{ sm }_{\xi }ʻʻ\nu &\qquad \text{(2)}\\ \vdash . (1).(2).\supset \vdash .\text{Prop} \end{array} \]

*119·64. \(\vdash \colon\ldotp \exists ! \text{ sm }_{\xi }ʻʻ\mu .\supset :\mu \geq \nu .\equiv . \exists !(\mu -_{c}\nu )_{\xi }\)

Dem. \[ \begin{array}{l} \vdash . *117·24. \supset \vdash \colon\ldotp \text{Hp}.\supset :\mu \geq \nu .&\supset .\mu ,\nu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\mu .\\ [*119·61] & \supset .(\mu -_{c}\nu )_{\xi } = (\text{ sm }_{\xi }ʻʻ\mu -_{c}\nu )_{\xi } &\qquad \text{(1)}\\ \vdash . *117·24·244 .\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu \geq \nu .&\supset .\text{ sm }_{\xi }ʻʻ\mu \geq \nu .\\ [*119·27] &\supset . \exists ! (\text{ sm }_{\xi }ʻʻ\mu -_{c}\nu )_{\xi }.\\ [(1)] &\supset . \exists !(\mu -_{c}\nu )_{\xi } &\qquad \text{(2)}\\ \vdash .(2).*119·26.\supset \vdash .\text{Prop} \end{array} \]

Summary of *120.

Inductive Cardinals are those that obey mathematical induction starting from 0, i.e. in the language of Part II, Section E, they are the posterity of 0 with respect to the relation of \(\nu\) to \(\nu+_{c}1\), or, in more popular language, they are those that can be reached from 0 by successive additions of 1. In former days, these were supposed to be all the cardinals, and mathematical induction was treated as a kind of self-evident axiom. We now know that only certain cardinals obey mathematical induction starting from 0. It is these cardinals which are to be considered in this number. They embrace 0, 1, 2, ... and generally all those cardinals which would be commonly called finite, all those which can be expressed in the usual Arabic system of numeration, and no others. The propositions to be proved concerning them in this number are elementary and familiar; the interest lies entirely in the definition and method of proof, not in the propositions themselves.

Put \(\text{NC induct} = \hat{\alpha} \{\alpha (+_{c}1)_{*}0\}\quad\text{Df}.\)

Since \((+_{c}1)_{*}\) has necessarily its domain and converse domain of the same type, it is important to be careful in noting the relations of type. Accordingly we also put \[ \text{N}_{\xi }\text{C induct} = \hat{\alpha} \{\alpha (+_{c}1)_{*}0_{\xi }\} \quad\text{Df}. \]

We begin by applying the propositions of *90. Thus we have

*120·11. \(\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi 0_{\eta }:\supset .\phi \alpha\)

*120·12. \(\vdash .0\in \text{NC induct}\)

*120·121. \(\vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}1)_{\xi }\in \text{N}_{\xi }\text{C induct}\)

*120·13. \(\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi 0_{\eta }:\supset .\phi \alpha \)

*120·15. \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\text{ sm }ʻʻ\alpha \in \text{NC induct}\)

*120·151. \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .a+_{c}1\in \text{NC induct}\)

*120·152. \(\vdash :\alpha \in \text{NC}.\text{ sm }ʻʻ\alpha \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)

We then proceed to deduce the elementary properties of inductive classes, putting \[ \text{Cls induct} = sʻ\text{NC induct}. \]

We have

[Pg 208]

*120·21. \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}\)

*120·211. \(\vdash :\text{Nc}ʻ\rho \in \text{NC induct}-\iota ʻ\Lambda .\supset .\rho \in \text{Cls induct}\)

(We do not have an equivalence here, because, for aught we know, it might be possible to determine the ambiguity of \(\text{Nc}ʻ\rho\) so that \(\text{Nc}ʻ\rho = \Lambda\), even when \(\rho \in \text{Cls induct}\). his will not be possible, however, if the axiom of infinity is assumed.)

*120·212·213. \(\vdash .\Lambda ,\iota ʻx\in \text{Cls induct}\)

*120·214. \(\vdash \colon\ldotp \rho \text{ sm }\sigma .\supset :\rho \in \text{Cls induct}.\equiv .\sigma \in \text{Cls induct}\)

We have a set of propositions applying induction to classes directly, and not through the intermediary of cardinals. Thus we have

*120·251. \(\vdash :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct}\)

*120·26. \(\vdash \colon\ldotp \rho \in \text{Cls induct}:\phi\eta .\supset _{\eta ,x}.\phi(\eta \cup \iota ʻx):\phi\Lambda :\supset .\phi\rho\)

We then state the axiom of infinity, and prove (*120·33) that it is equivalent to the assumption that if \(\alpha\) is an inductive cardinal, \(\alpha \neq \alpha +_{c}1\). To prove this, we first prove various propositions about \(\alpha +_{c}1\), among others the following:

*120·311. \(\vdash :\exists !\alpha +_{c}1.a+_{c}1=\beta +_{c}1.\supset .\alpha =\text{ sm }ʻʻ\beta .\exists !\alpha\)

*120·322. \(\vdash \colon\ldotp \alpha \in \text{NC induct}.\supset :\exists !\alpha .\equiv .\alpha \neq \alpha +_{c}1\)

We then proceed to consider subtraction (*120·41—·418), which only gives a cardinal number when the subtrahend is an inductive cardinal. We have

*120·41. \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\supset :\alpha +_{c}\nu =\beta +_{c}\nu .\supset .\alpha =\text{ sm }ʻʻ\beta\)

We might validly put \(\alpha = \beta\) instead of \(\alpha =\text{ sm }ʻʻ\beta\), since \(\alpha = \beta\) will be true whenever it is significant.

We have

*120·411. \[\begin{align}\vdash \colon\ldotp \nu \in &\text{NC induct}.\supset :\\ &\exists !\gamma -_{c}\nu .\supset .\gamma -_{c}\nu \in \text{N}_{0}\text{C}:\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \in \text{N}_{0}\text{C}\end{align}\]

*120·4111. \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset :\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )_{\xi }\in \text{N}_{0}\text{C}\)

Hence we arrive at the conditions requisite for the usual point of view of subtraction; namely,

*120·412. \(\vdash :\nu \in \text{NC induct}.\gamma \geq \nu .\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset .(\gamma -_{c}\nu )_\xi =\{(\iota \alpha )(\alpha +_{c}\nu =\gamma )\}_{\xi }\)

Also from *120·4111 we deduce

*120·414. \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .(\mu -_{c}1)_{\xi }\in \text{N}_{0}\text{C}\)

And from *120·411.*119·34, we find

*120·416. \(\vdash :\nu \in \text{NC induct}.\exists !\gamma -_{c}\nu .\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma\)

We prove next that no proper part of an inductive class is similar to the whole (*120·426), i.e. that inductive classes are non-reflexive, and various connected propositions, e.g.

[Pg 209]

*120·423. \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct} -\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta }\)

*120·4232. \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct} -\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct} -\iota ʻ\Lambda .\alpha =(\beta +_{c}1)_{\eta }\)

*120·428. \(\vdash :\nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\alpha \neq 0.\supset .\alpha +_{c}\nu >\nu\)

*120·429. \(\vdash \colon\ldotp \nu \in \text{NC induct}.\supset :\mu >\nu .\equiv .\mu \geq \nu +_{c}1\)

The last two of the above propositions do not hold in general when \(\nu\) is a cardinal which is not inductive.

We prove next that if \(\alpha\) is an existent inductive cardinal, then any existent cardinal is greater than, equal to, or less than \(\alpha\) (*120·441); that if \(\alpha\), \(\beta\) are inductive cardinals, so is \(\alpha +_{c}\beta\) (*120·45·4501), and if \(\alpha+_{c}\beta\) is an inductive cardinal other than \(\Lambda\), so are \(\alpha\) and \(\beta\) (*120·452). We then have some propositions dealing with mathematical induction starting from 1 or 2, e.g.

*120·4622. \[\begin{align}\vdash \colon\ldotp \alpha \in \text{NC}.\beta \in &\text{NC}(\eta ).\exists !\text{ sm }_{\xi }ʻʻ\beta .\supset :\\ &\beta (+_{c} 1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\equiv .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha\end{align}\]

*120·47. \(\vdash \colon\colon \beta \in \text{N}_{\eta }\text{C induct} -\iota ʻ0.\equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \mu :1_{\eta }\in \mu :\supset _{\mu }.\beta \in \mu\)

From *120·452 we deduce

*120·48. \(\vdash :\beta \in \text{NC induct}.\beta \geq \alpha .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)

so that any number less than an inductive number is inductive. Hence

*120·481. \(\vdash :\eta \in \text{Cls induct}.\xi \subset \eta .\supset .\xi \in \text{Cls induct}\)

which is a proposition constantly used, and

*120·491. \(\vdash \colon\ldotp \xi {\sim}\in \text{Cls induct}.\equiv :\beta \in \text{NC induct}.\supset _{\beta }.\exists !\beta \cap \text{Cl}ʻ\xi\)

We then prove that if \(\alpha\), \(\beta\) are inductive cardinals, \(\alpha \times _{c}\beta\) and \(\alpha ^\beta\) are either inductive cardinals or \(\Lambda\) (*120·5*120·52), while conversely if \(\alpha\times _{c} \beta\) or \(\alpha ^\beta\) is an existent inductive cardinal, \(\alpha\) and \(\beta\) are so also, with exceptions for 0 and 1 (*120·512·56·561). Hence we infer the uniqueness of division and the taking of roots (*120·51·53·55) so long as inductive numbers are concerned.

We have next a set of propositions on the axiom of infinity and the multiplicative axiom. We prove (*120·61) that if there is any existent cardinal which is not inductive, the axiom of infinity is true. From *83·9·904, we infer by induction that if \(\kappa\) is an inductive class of which \(\Lambda\) is not a number, \({\in}_{\Delta}ʻ\kappa\) exists (*120·62), whence it follows that either the multiplicative axiom or the axiom of infinity must be true (*120·64).

Finally, we have a set of propositions on inductive classes. We prove

*120·71. \(\vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \cup \sigma \in \text{Cls induct}.\equiv .\rho +\sigma \in \text{Cls induct}\)

*120·74. \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{Cl}ʻ\rho \in \text{Cls induct}\)

*120·75. \(\vdash :sʻ\kappa \in \text{Cls induct}.\equiv .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\)

with analogous propositions (involving however a hypothesis as to [Pg 210]\(\kappa\)) on the subject of \({\in}_{\Delta}ʻ\kappa\).

The propositions of the present number are essential to the ordinary arithmetic of finite numbers. In the present work, however, they are not much used after the present section until we reach Part V, Section E, where we deal with the ordinal theory of finite and infinite.

*120·01. \(\text{NC induct} = \hat{\alpha}\{\alpha (+_{c}1)_{*}0\} \quad\text{Df}\)

Note that in virtue of our general conventions for descriptive functions of two arguments (*38), \[ +_{c}1=\hat{\alpha} \hat{\beta} (a=\beta +_{c}1). \] That is, \(+_{c}1\) is the relation of a cardinal to its immediate predecessor. It is the number written in the usual mathematical notation as +1 in the series of positive and negative integers, just as its converse is the number -1. (It should be observed that if \(\nu\) is any cardinal, +\(\nu\) is not identical with \(\nu\), since +\(\nu\) is a relation, while \(\nu\) is a class of classes.)

*120·011. \(\text{N}_{\xi }\text{C induct} =\hat{\alpha} \{\alpha (+_{c}1)_{*}0_{\xi }\} \quad\text{Df}\)

All members of \(\text{N}_{\xi }\text{C induct}\) belong to the same type as \(0_{\xi }\), so that, if \(\alpha\) is any member of \(\text{N}_{\xi }\text{C induct}\), "\(\xi \in \alpha\)" is significant.

*120·02. \(\text{Cls induct} = sʻ\text{NC induct} \quad\text{Df}\)

*120·021. \(\text{Cls}_{\xi }\text{induct} = sʻ\text{N}_{\xi }\text{C induct} \quad\text{Df}\)

In virtue of these definitions an inductive class is one whose cardinal is an inductive cardinal.

*120·03. \(\text{Infin ax}.=:\alpha \in \text{NC induct}.\supset _{a}.\exists !\alpha \quad\text{Df}\)

"\(\text{Infin ax}\)," like "\(\text{Mult ax}\)," is an arithmetical hypothesis which some will consider self-evident, but which we prefer to keep as a hypothesis, and to adduce in that form whenever it is relevant. Like "\(\text{Mult ax}\)," it states an existence-theorem. In the above form, it states that, if \(\alpha\) is any inductive cardinal, there is at least one class (of the type in question) which has \(\alpha\) terms. An equivalent assumption would be that, if \(\rho\) is any inductive class, there are objects which are not members of \(\rho\). For in that case, if \(x\) be such an object, \(\text{Nc}ʻ(\rho \cup \iota ʻx)=\text{Nc}ʻ\rho +_{c}1\). Hence by induction, every inductive cardinal must exist. Another equivalent assumption would be that \(\text{V}\) (the class of all objects of the type in question) is not an inductive class. The assumption that \(\aleph _{0}\) exists in the type in question is, as we shall see, a stronger assumption than the above, unless we assume the multiplicative axiom.

If the axiom of infinity is true, the inductive cardinals are all different one from another, i.e. \(\alpha +_{c}\beta\), where \(\alpha\) and \(\beta\) are inductive cardinals, is not equal to \(\alpha\) unless \(\beta = 0\). But if the axiom of infinity is false, then, in any assigned type, all the cardinals after a certain one are \(\Lambda\). (Except in the lowest type, the last existent cardinal must be a power of 2.) That is, if (say) 8 were the largest existent cardinal in the type in question, we should[Pg 211] have, in that type, \(9 = \Lambda\), and the same would hold of 10, 11, .... This possibility has to be taken account of in what follows.

In order to give typical definiteness to the axiom of infinity, we write

*120·04. \(\text{Infin ax}(x).=:\alpha \in \text{NC induct}.\supset _{a}.\exists !\alpha (x) \quad\text{Df}\)

Then "\(\text{Infin ax} (x)\)" states that, if \(\alpha\) is any inductive cardinal, there are at least \(\alpha\) objects of the same type as \(x\).

*120·1. \(\vdash :\alpha \in \text{NC induct}.\equiv .\alpha (+_{c}1)_{*}0 \quad[(*120·01)]\)

*120·101. \[\begin{align}&\vdash \colon\colon \alpha \in \text{NC induct}. \equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.\xi +_{c}1\in \mu :0\in \mu :\supset _{\mu }.\alpha \in \mu \\ &[*120·1.*90·131.*38·12]\end{align}\]

The right-hand side of the above equivalence gives the usual formula for mathematical induction. Observe that the conditions of significance require that \(\xi +_{c}1\) should be taken in the same type as \(\xi\). This fact is specially relevant in the proof of *120·15.

The symbol "\(\text{NC induct}\)" is of ambiguous type not necessarily the same in different occurrences; also, according to the convention explained in the prefatory statement as holding for \(\text{NC}\) and \(\text{NC induct}\), "\(\alpha ,\beta \in \text{NC induct}\)" will not imply that \(\alpha\) and \(\beta\) are of the same type. Accordingly to avoid error in connection with *120·1·101 typical definiteness is required as in the three following propositions.

*120·102. \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct}. \equiv .\alpha (+_{c} 1)_{*}0_{\eta } \quad[(*120·011)]\)

*120·103. \[\begin{align}&\vdash \colon\colon \alpha \in \text{N}_{\eta }\text{C induct}. \equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \mu :0_{\eta }\in \mu :\supset _{\mu }.\alpha \in \mu \\ &[*120·101]\end{align}\]

*120·11. \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\phi\xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi 0_{\eta }:\supset .\phi \alpha \\ &[*120·102.*90·112]\end{align}\]

*120·12. \(\vdash .0\in \text{NC induct} \quad\left[*120·101 \frac{0}{\alpha}\right]\)

*120·121. \(\vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}1)_{\xi }\in \text{N}_{\xi }\text{C induct} \quad[*90·172.*120·102]\)

By means of this proposition and *120·12, any assigned cardinal in the series of natural numbers can be shown to be an inductive cardinal; thus e.g. to show that 27 is an inductive cardinal, we shall only have to use *120·121 twenty-seven times in succession.

*120·122. \(\vdash .1\in \text{NC induct} \quad[*120·12·121.*110·641]\)

*120·123. \(\vdash .2\in \text{NC induct}. \text{etc}. \quad[*120·122·121.*110·643]\)

*120·124. \(\vdash .\alpha +_{c}1\neq 0\)

Dem. \[ \begin{array}{l} \vdash .*110·4. \text{Transp}.&\supset \vdash :\alpha {\sim}\in \text{NC}.\supset .\alpha +_{c}1=\Lambda .\\ [*101·12] &\supset .\alpha +_{c}1\neq 0 &\qquad \text{(1)}\\ \vdash .*110·632. &\supset \vdash \colon\ldotp \alpha \in \text{NC}.\supset :\xi \in \alpha +_{c}1.\supset .\exists !\xi :\\ [*24·63] &\supset :\Lambda {\sim}\in \alpha +_{c}1:\\ [*54·102] &\supset :\alpha +_{c}1\neq 0 &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 212]

*120·13. \(\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi 0_{\eta }:\supset .\phi \alpha\)

Dem. \[ \begin{array}{l} \vdash .*120·121. &\supset \vdash \colon\ldotp \xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\supset :\\ &\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi (\xi +_{c}1) &\qquad \text{(1)}\\ \vdash .*120·12. &\supset \vdash :\phi 0_{\eta }.\supset .0_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi 0_{\eta } &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ \xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .&\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi (\xi +_{c}1):0_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi 0_{\eta }:\\ \left[*120·11 \frac{\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi}{\phi \xi }\right]&\supset :\alpha \in \text{N}_{\eta }\text{C induct}.\phi \alpha \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

The above proposition is often convenient for inductive proofs.

*120·14. \(\vdash .\text{NC induct} \subset \text{NC}\)

Dem. \[ \begin{array}{l} \vdash . *110·42. \text{Simp}.\supset \vdash :\alpha \in \text{NC}.\supset .\alpha +_{c}1\in \text{NC} &\qquad \text{(1)}\\ \vdash .(1).*101·11. *120·11 \frac{\alpha \in \text{NC}}{\phi \alpha} .\supset \vdash .\text{Prop} \end{array} \]

This proposition does not show that every inductive cardinal is an existent cardinal; to obtain this, we require the axiom of infinity.

*120·15. \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\text{ sm }ʻʻ\alpha \in \text{NC induct}\)

i.e. a cardinal which is not null and is inductive in any one type is also inductive in any other type.

Dem. \[ \begin{array}{l} \vdash .*101·15.*120·12.&\supset \vdash .\text{ sm }_{\eta }ʻʻ0_{\xi }\in \text{N}_{\eta }\text{C induct} &\qquad \text{(1)}\\ \vdash .*110·4. &\supset \vdash .\alpha =\Lambda _{\xi }.\supset .(\alpha +_{c}1)_{\xi } = \Lambda _{\xi } &\qquad \text{(2)}\\ \vdash .*118·201. &\supset \vdash :\exists !(\alpha +_{c}1)_{\xi }.\supset .\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }=(\alpha +_{c}1)_{\eta }\\ [*118·241.*110·4] &= (\text{ sm }_{\eta }ʻʻ\alpha +_{c} 1)_{\eta } &\qquad \text{(3)}\\ \vdash .*120·121. &\supset \vdash :\exists !(\alpha +_{c}1)_{\xi }.\text{ sm }_{\eta }ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}.\supset .(\text{ sm }_{\eta }ʻʻ\alpha +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\\ [(3)] &\supset .\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }\in 1\,\text{N}_{C}\text{induct} &\qquad \text{(4)}\\ \vdash .(4).*2·2. & \supset \vdash \colon\ldotp \text{ sm }_{\eta }ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}.\supset :\\\\ &(\alpha +_{c}1)_{\xi }=\Lambda _{\xi }.\lor.\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }\in \text{N}_{\eta }\text{C induct} &\qquad \text{(5)}\\ \vdash .(2).(5).*3·48.&\supset \vdash \colon\ldotp \alpha =\Lambda _{\xi }.\lor.\text{ sm }_{\eta }ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}:\supset :\\ &(\alpha +_{c}1)_{\xi }=\Lambda _{\xi }.\lor.\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }\in \text{N}_{\eta }\text{C induct} &\qquad \text{(6)}\\ \vdash .(1).(6).*120·11.*4·6.\supset \vdash .\text{Prop} \end{array} \]

*120·151. \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\alpha +_{c}1\in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·15. \supset \vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\exists !\alpha .&\supset .\text{ sm }_\eta ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}.\\ [*120·121] &\supset .(\text{ sm }_{\eta }ʻʻ\alpha +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\\ [*118·241.*120·14] & \supset .(\alpha +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}:\supset \vdash .\text{Prop} \end{array} \]

*120·152. \(\vdash :\alpha \in \text{NC}.\text{ sm }ʻʻ\alpha \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*100·521. \supset \vdash :\text{Hp}.&\supset .\text{ sm }ʻʻ\text{ sm }ʻʻ\alpha =\alpha .\\ [*120·15] &\supset .\alpha \in \text{NC induct} &\qquad \text{(1)}\\ \vdash .*37·29. &\supset \vdash :\text{Hp}.\supset .\exists !\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

[Pg 213]

The following propositions, giving alternative forms for the definition of inductive classes, are inserted in order to show that the theory of inductive classes might be treated in a less arithmetical manner than we have adopted.

*120·2. \(\vdash :\rho \in \text{Cls induct}.\equiv .(\exists \alpha ).\alpha \in \text{NC induct}.\rho \in \alpha \quad[(*120·02)]\)

*120·201. \(\vdash \colon\ldotp \rho \text{ sm }\sigma .\supset :\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}.\equiv .\text{N}_{0}\text{c}ʻ\sigma \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*100·35.*103·13.*100·511.\supset \\ \vdash :\text{Hp}.\supset .\text{N}_{0}\text{c}ʻ\rho =\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\sigma .\text{N}_{0}\text{c}ʻ\sigma =\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\rho :\\ [*120·152.*103·13] \supset \vdash .\text{Prop} \end{array} \]

*120·21. \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·14·2. \supset \vdash :\rho \in \text{Cls induct}.&\equiv .(\exists \alpha ).\alpha \in \text{NC induct}.\alpha \in \text{NC}.\rho \in \alpha .\\ [*103·27] &\equiv .(\exists \alpha ).\alpha \in \text{NC induct}. \text{N}_{0}\text{c}ʻ\rho =\alpha .\\ [*13·195] &\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}:\supset \vdash .\text{Prop} \end{array} \]

Note that "\(\rho \in \text{Cls induct}.\equiv .\text{Nc}ʻ\rho \in\text{NC induct}\)" is not proved above. The proof encounters the difficulty that we may have \(\text{Nc}ʻ\rho = \Lambda\); in order to establish our proposition in this case, we have to show that if \(\Lambda \in \text{NC induct}\), then every class is an inductive class. We can however prove the following implication.

*120·211. \(\vdash :\text{Nc}ʻ\rho \in \text{NC induct} - \iota ʻ\Lambda .\supset .\rho \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*100·511.&\supset \vdash :\text{Hp}.\supset.\text{ sm }ʻʻ\text{Nc}ʻp=\text{N}_{0}\text{c}ʻ\rho .\\ [*120·15] &\supset . \text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}.\\ [*120·21] &\supset .\rho \in \text{Cls induct}:\supset \vdash .\text{Prop} \end{array} \]

*120·212. \(\vdash .\Lambda \in \text{Cls induct} \quad[*120·211·12]\)

*120·213. \(\vdash .\iota ʻ\alpha \in \text{Cls induct} \quad [*120·211·122]\)

*120·214. \(\vdash \colon\ldotp \rho \text{ sm }\sigma .\supset :\rho \in \text{Cls induct}.\equiv .\sigma \in \text{Cls induct} \quad[*120·201·21]\)

The following propositions are lemmas for *120·24.

*120·22. \(\vdash \colon\colon \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset _{\mu }.\rho \in \mu \colon\ldotp \supset .\rho \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·212. &\supset \vdash .\Lambda \in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .*51·2. &\supset \vdash \colon\ldotp y\in \eta .\supset :\eta \cup \iota ʻy=\eta :\\ [*13·12] &\supset :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .*110·63. &\supset \vdash \colon\ldotp y{\sim}\in \eta .\supset :\text{Nc}ʻ(\eta \cup \iota ʻy)=\text{Nc}ʻ\eta +_{c}\\ [(*110·03)] &= \text{N}_{0}\text{c}ʻ\eta +_{c}1:\\ [*120·121] & \supset :\text{N}_{0}\text{c}ʻ\eta \in \text{NC induct}.\supset .\text{N}_{0}\text{c}ʻ(\eta \cup \iota ʻy)\in \text{NC induct}:\\ [*120·21·211] & \supset :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct} &\qquad \text{(3)}\\ \vdash .(2).(3). &\supset \vdash :\eta \in \text{Cls induct}. \supset . \eta \cup \iota ʻy\in \text{Cls induct} &\qquad \text{(4)}\\ \vdash .*10·1.(1).(4).\supset \vdash .\text{Prop} \end{array} \]

[Pg 214]

*120·221. \(\vdash \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\text{Nc}ʻ\rho \subset \mu :\supset .\text{Nc}ʻ\rho +_{c}1\subset \mu \)

Dem. \[ \begin{array}{l} \vdash .*110·63.*100·31.\supset \\ \vdash :\zeta \in \text{Nc}ʻ\rho +_{c}1.&\equiv .(\exists \eta ,y).\eta \in \text{Nc}ʻ\rho .y{\sim}\in \eta .\zeta =\eta \cup \iota ʻy &\qquad \text{(1)}\\ \vdash .*22·1. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\eta \in \text{Nc}ʻ\rho .\supset .\eta \in \mu .\\ [*10·1] & \supset .\eta \cup \iota ʻy\in \mu :\\ [*3·41] & \supset :\eta \in \text{Nc}ʻ\rho .y{\sim}\in \eta .\supset .\eta \cup \iota ʻy\in \mu :\\ [*13·12] &\supset :\eta \in \text{Nc}ʻ\rho .y{\sim}\in \eta .\zeta =\eta \cup \iota ʻy.\supset .\zeta \in \mu &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\zeta \in \text{Nc}ʻ\rho +_{c}1.\supset .\zeta \in \mu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·222. \(\vdash \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\xi \in \text{NC}.\xi \subset \mu :\supset .\xi +_{c}1\subset \mu\)

Dem. \[ \begin{array}{l} \vdash .*100·4.\supset \vdash :\text{Hp}.\exists !\xi .&\supset .(\exists \alpha ).\xi =\text{Nc}(\zeta )ʻ\alpha .\text{Nc}(\zeta )ʻ\alpha \subset \mu .\\ [*120·221] &\supset .(\exists \alpha ).\xi =\text{Nc}(\zeta )ʻ\alpha .\text{Nc}ʻ\alpha +_{c}1\subset \mu .\\ [*118·01] &\supset .\xi +_{c}1\subset \mu &\qquad \text{(1)}\\ \vdash .*110·4.&\supset \vdash :{\sim}\exists !\xi .\supset .\xi +_{c}1\subset \mu &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

The proof of this proposition might also proceed by the use of uniform formal numbers, employing *118·241.

*120·23. \(\vdash \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset .\text{Cls induct}\subset \mu\)

Dem. \[ \begin{array}{l} \vdash .*51·2.*54·1. &\supset \vdash :\text{Hp}.\supset .0\subset \mu &\qquad \text{(1)}\\ \vdash .*120·222·14. & \supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \text{NC induct}.\xi \subset \mu .\supset _{\xi }.\xi +_{c}1\subset \mu &\qquad \text{(2)}\\ \vdash .(1).(2).*120·13. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \text{NC induct}.\supset .\xi \subset \mu :\\ [*40·151.(*120·02)] &\supset :\text{Cls induct}\subset \mu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·24. \(\vdash \colon\colon \rho \in \text{Cls induct}.\equiv \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset _{\mu }.\rho \in \mu\)

Dem. \[ \begin{array}{l} \vdash .*120·23.\supset \vdash \colon\colon \rho \in \text{Cls induct}.\supset \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\mu \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset .\rho \in \mu &\qquad \text{(1)}\\ \vdash .(1).*120·22.\supset \vdash .\text{Prop} \end{array} \]

This proposition might be used to define inductive classes. It gives a form of mathematical induction applicable to classes instead of to numbers. Virtually it states that an inductive class is one which can be formed by adding members one at a time, starting from \(\Lambda\). This is made more explicit in *120·25. Instead of \(\eta\in \mu .\supset _{\eta ,y}.\mu \cup \iota ʻy\in \mu\), in the above propositions, as well as in those that follow, we may plainly substitute \[ \eta \in \mu .y{\sim}\in \eta .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu . \]

*120·25. \[\begin{align}&\vdash :M=\hat{\eta} \hat{\zeta}\{(\exists y).\zeta =\eta \cup \iota ʻy\}.\supset .\text{Cls induct}=\overleftarrow{M}_{*}ʻ\Lambda\\ &[*120·24.*90·131]\end{align}\]

*120·251. \(\vdash :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct} \quad[*90·172.*120·25]\)

*120*26. \[\begin{align}&\vdash \colon\ldotp \rho \in \text{Cls induct}:\phi\eta .\supset _{\eta ,x}.\phi(\eta \cup \iota ʻx):\phi\Lambda :\supset \phi\rho \\ &[*120·25.*90·112]\end{align}\]

[Pg 215]

*120·261. \[\begin{align}&\vdash \colon\ldotp \rho \in \text{Cls induct}:\eta \in \text{Cls induct}.\phi\eta .\supset _{\eta ,x}.\phi(\eta \cup \iota ʻx):\phi\Lambda :\supset .\phi\rho \\ &[*120·26·251·212]\end{align}\]

*120·27. \(\vdash :\rho \in \text{Cls induct}.\supset .\text{Nc}ʻ\rho \cap tʻ\gamma \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·12. &\supset \vdash .\text{Nc}ʻ\Lambda \cap tʻ\gamma \in \text{NC induct} &\qquad \text{(1)}\\ \vdash .*13·12. &\supset \vdash :\text{Nc}ʻ\eta \cap tʻ\gamma \in \text{NC induct}.y\in \eta .\supset .\\ &\text{Nc}ʻ(\eta \cup \iota ʻy)\cap tʻ\gamma \in \text{NC induct} &\qquad \text{(2)}\\ \vdash .*110·63.*120·121.\supset\\ &\vdash :\text{Nc}ʻ\eta \cap tʻ\gamma \in \text{NC induct}.y{\sim}\in \eta .\supset .\text{Nc}ʻ(\eta \cup \iota ʻy)\cap tʻ\gamma \in \text{NC induct} &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*120·26.\supset \vdash .\text{Prop} \end{array} \]

This proposition also follows immediately from *12·21·15.

*120·3. \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha \quad[(*120·03)]\)

*120·301. \(\vdash \colon\ldotp \text{Infin ax}(x).\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha (x) \quad[(*120·04)]\)

*120·31. \(\vdash :\exists !\text{Nc}ʻ\alpha +_{c}1.\text{Nc}ʻ\alpha +_{c}1=\text{Nc}ʻ\beta +_{c}1.\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\alpha \text{ sm }\beta \)

Dem. \[ \begin{array}{l} \vdash .*110·63. &\supset \vdash \colon\ldotp \text{Nc}ʻ\alpha +_{c}1=\text{Nc}ʻ\beta +_{c}1.\equiv :\\ &(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy.\equiv _{\xi }.(\exists \delta ,z).\delta \text{ sm }\beta .z{\sim}\in \delta .\xi =\delta \cup \iota ʻz:\\ [*10·1] &\supset :\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\supset .(\exists \delta ,z).\delta \text{ sm }\beta .z{\sim}\in \delta .\gamma \cup \iota ʻy=\delta \cup \iota ʻz.\\ [*73·72·3] &\supset .(\exists \delta ).\delta \text{ sm }\beta .\gamma \text{ sm }\delta .\\ [*73·32] &\supset .\gamma \text{ sm }\beta .\\ [*73·32] &\supset .\alpha \text{ sm }\beta &\qquad \text{(1)}\\ \vdash .*110·63. &\supset \vdash :\text{Hp}.\supset .(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma &\qquad \text{(2)}\\ \vdash .(1).(2).*100·321.\supset \vdash .\text{Prop} \end{array} \]

*120·311. \[\begin{align}&\vdash :\exists !\alpha +_{c}1.\alpha +_{c}1=\beta +_{c}1.\supset .\alpha =\text{ sm }ʻʻ\beta .\exists !\alpha\\ &[*120·31.*110·4.*103·16·4·2]\end{align}\]

*120·32. \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\alpha \neq \alpha +_{c}1\)

Dem. \[ \begin{array}{l} \vdash .*101·22.*110·641.&\supset \vdash .0_{\xi }\neq 0_{\xi }+_{c}1 &\qquad \text{(1)}\\ \vdash .*120·311.*110·44.&\supset \vdash :\alpha \in \text{NC}.\exists !\alpha +_{c}1.\alpha +_{c}1=\alpha +_{c}1+_{c}1.\supset .\alpha =\alpha +_{c}1:\\ [\text{Transp}] &\supset \vdash :\alpha \in \text{NC}.\exists !\alpha +_{c}1.\alpha \neq \alpha +_{c}1.\supset .\alpha +_{c}1\neq \alpha +_{c}1+_{c}1:\\ [*118·2·25] &\supset \vdash :\alpha \in \text{NC}(\xi ).\exists !(\alpha +_{c}1)_{\xi }.\alpha \neq (\alpha +_{c}1)_{\xi }.\supset .(\alpha +_{c}1)_{\xi }\neq \{(\alpha +_{c}1)_{\xi }+_{c}1\}_{\xi} &\qquad \text{(2)}\\ \vdash .(2).&\supset \vdash \colon\ldotp \alpha \in \text{NC}(\xi ).\alpha \neq (\alpha +_{c}1)_{\xi }.\supset :(\alpha +_{c}1)_{\xi }=\Lambda .\lor.(\alpha +_{c}1)_{\xi }\neq \{(\alpha +_{c}1)_{\xi }+_{c}1\}_{\xi} &\qquad \text{(3)}\\ \vdash .*110·4. \text{Transp}.&\supset \vdash \colon\ldotp \alpha {\sim}\in \text{NC}(\xi ).\lor.\alpha =\Lambda _{\xi }:\supset .(\alpha +_{c}1)_{\xi }=\Lambda _{\xi } &\qquad \text{(4)}\\ \vdash .(3).(4).&\supset \vdash \colon\ldotp \alpha =\Lambda _{\xi }.\lor.\alpha \neq (\alpha +_{c}1)_{\xi }:\supset :\\ &(\alpha +_{c}1)_{\xi }=\Lambda _{\xi }.\lor.(\alpha+_{c}1)_{\xi }\neq {(\alpha +_{c}1)_{\xi }+_{c}1}_{\xi } &\qquad \text{(5)}\\ \vdash .(1).(5).*120·11.&\supset \vdash \colon\ldotp \alpha \in \text{N}_{\xi }\text{C induct}.\supset :\alpha =\Lambda _{\xi }.\lor.\alpha \neq (\alpha +_{c}1)_{\xi }\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

[Pg 216]

*120·321. \(\vdash :\alpha \neq \alpha +_{c}1.\supset .\exists !\alpha\)

Dem. \[ \begin{array}{l} \vdash .*110·4. \text{Transp}.&\supset \vdash :\alpha =\Lambda .\supset .\alpha +_{c}1=\Lambda &\qquad \text{(1)}\\ \vdash .(1).\text{Transp}. &\supset \vdash .\text{Prop} \end{array} \]

*120·322. \(\vdash \colon\ldotp \alpha \in \text{NC induct}.\supset :\exists !\alpha .\equiv .\alpha \neq \alpha +_{c}1 \quad[*120·32·321]\)

*120·33. \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\alpha \neq \alpha +_{c}1 \quad[*120·3·322]\)

*120·41. \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\supset :\alpha +_{c}\nu =\beta +_{c}\nu .\supset .\alpha =\text{ sm }ʻʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*110·4. \text{Transp}.*118·25.&\supset \vdash :(\alpha +_{c}\nu )_{\xi }=\Lambda .\supset .\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi }=\Lambda &\qquad \text{(1)}\\ \vdash .*118·25 .&\supset \vdash \colon\colon \exists !{\alpha +_{c}(\nu +_{c}1)_{\xi }}_{\xi }.\supset \colon\ldotp \exists !\{(\alpha +_{c}\nu )_{\xi }+_{c}1\}_{\xi}\colon\ldotp \\ [*120·311.*110·4.*118·201] &\supset \colon\ldotp \{(\alpha +_{c}\nu )_{\xi }+_{c}1\}_{\xi}=\{(\beta +_{c}\nu )_{\xi }+_{c}1\}_{\xi }.\supset .(\alpha +_{c}\nu )_{\xi}=(\beta +_{c}\nu )_{\xi }\colon\ldotp \\ [\text{Syll}.*118·25] &\supset \colon\ldotp (\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta :\supset :\\ &\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}=\{\beta +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(2)}\\ \vdash .(2).\text{Comm}.&\supset \vdash \colon\colon (\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta :\supset \colon\ldotp \\ &\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi} = \Lambda :\lor:\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}=\{\beta +_{c}(\nu +_{c}1)_{\xi }\}_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(3)}\\ \vdash .(1).(3).&\supset \vdash \colon\colon (\alpha +_{c}\nu )_{\xi }=\Lambda :\lor:(\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta \colon\ldotp \supset \colon\ldotp \\ &\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}=\Lambda :\lor:\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi }=\{\beta +_{c}(\nu +_{c}1)_{\xi }\}_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(4)}\\ \vdash .*110·4.*118·21.&\supset \vdash \colon\ldotp \exists !(\beta +_{c}0)_{\xi }.\supset :\beta \in \text{NC}.\exists !\text{ sm }_{\xi }ʻʻ\beta :\\ [*102·87.*100·51] &\supset :\text{ sm }_{\xi }ʻʻ\alpha =\text{ sm }_{\xi }ʻʻ\beta .\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(5)}\\ \vdash .*110·6·4.&\supset \vdash :\exists !(\alpha +_{c}0)_{\xi }.(\alpha +_{c}0)_{\xi }=(\beta +_{c}0)_{\xi }.\supset .\text{ sm }_{\xi }ʻʻ\alpha =\text{ sm }_{\xi }ʻʻ\beta .\\ [(5)] &\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(6)}\\ \vdash .(6).\text{Exp}.*4·6.&\supset \vdash \colon\ldotp (\alpha +_{c}0)_{\xi }=\Lambda :\lor:(\alpha +_{c}0)_{\xi }=(\beta +_{c}0)_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(7)}\\ \vdash .(4).(7).*120·11.\supset \\ \vdash \colon\colon \nu \in \text{N}_{\xi }\text{C induct}.&\supset \colon\ldotp (\alpha +_{c}\nu )_{\xi }=\Lambda :\lor:(\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(8)}\\) \vdash .*110·4. &\supset \vdash :\nu =\Lambda_\eta.\supset .(\alpha +_c{\nu} )_\xi=\Lambda &\qquad \text{(9)}\\ \vdash .*120·15.&\supset \vdash \colon\colon \nu \in \text{N}_\eta \text{C induct}-\iota ʻ\Lambda .\supset \colon\ldotp \text{ sm }_\xiʻʻ\nu =\text{N}_\xi\text{C induct}\colon\ldotp \\ [(8)] &\supset \colon\ldotp (\alpha +_c\text{ sm }_\xiʻʻ\nu )_\xi=\Lambda :\lor:(\alpha +_c\text{ sm }_\xiʻʻ\nu )_\xi=(\beta +_c\text{ sm }_\xiʻʻ\nu )_\xi.\supset .\alpha =\text{ sm }ʻʻ\beta \colon\ldotp \\ [*118·24] &\supset \colon\ldotp (\alpha +_c\nu )_\xi=\Lambda :\lor:(\alpha +_c\nu )_\xi=(\beta +_c\nu )_\xi.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(10)}\\ \vdash .(9).(10).\supset \vdash .\text{Prop} \end{array} \]

[Pg 217]

The above proposition establishes (with the natural limitations) the uniqueness (within each type) of subtraction (conceived as in *120·412) when the subtrahend is an inductive cardinal. (When the subtrahend is a non-inductive cardinal, subtraction ceases to give a unique result.) Hence we are led to the following extensions of *118 for the case of inductive cardinals:

*120·411. \[\begin{align}\vdash \colon\ldotp \nu \in &\text{NC induct}.\supset :\\ &\exists !\gamma -_{c}\nu .\supset .\gamma -_{c}\nu \in \text{N}_{0}\text{C}:\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \in \text{N}_{0}\text{C}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*119·1. \supset \vdash \colon\ldotp &\nu \in \text{NC induct}.\supset :\\ &\xi ,\eta \in \gamma -_{c}\nu .\supset .\text{Nc}ʻ\xi +_{c}\nu =\gamma .\text{Nc}ʻ\eta +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu .\\ [*20·22] &\supset .\text{Nc}ʻ\xi +_{c}\nu =\text{Nc}ʻ\eta +_{c}\nu .\exists !\text{Nc}ʻ\xi +_{c}\nu .\\ [*120·41.*100·511.(*110·03)] &\supset .\text{Nc}ʻ\xi =\text{Nc}ʻ\eta &\qquad \text{(1)}\\ \vdash .(1).*119·14. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !\gamma -_{c}\nu .\supset .\gamma -_{c}\nu \in \text{N}_{0}\text{C} &\qquad \text{(2)}\\ \vdash .*119·27.(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \geq \nu .\supset .(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \in \text{N}_{0}\text{C} &\qquad \text{(3)}\\ \vdash .*103·22.*119·27. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \in \text{N}_{0}\text{C}.\supset .\gamma \geq \nu &\qquad \text{(4)}\\ \vdash .(2).(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*120·4111. \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset :\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )_{\xi }\in \text{N}_{0}\text{C}\)

Dem. \[ \begin{array}{l} \vdash .*119·64. \supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \geq \nu .&\supset .\exists !(\gamma -_{c}\nu )_{\xi }.\\ [*120·411] &\supset .(\gamma -_{c}\nu )_{\xi }\in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ \vdash .(1).*119·26.*103·13.\supset \vdash .\text{Prop} \end{array} \]

*120·412. \(\vdash :\nu \in \text{NC induct}.\gamma \geq \nu .\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset .(\gamma -_{c}\nu )_{\xi }=\{({℩}\alpha )(\alpha +_{c}\nu =\gamma )\}_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*120·4111.\supset \vdash \colon\ldotp \text{Hp}.&\supset .(\gamma -_{c}\nu )_{\xi }\in \text{N}_{0}\text{C}.\\ [*119·34] &\supset .(\gamma -_{c}\nu )_{\xi }+_{c}\nu = \gamma &\qquad \text{(1)}\\ \vdash .*120·41.*103·43.*37·29.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha +_{c}\nu =\gamma .\beta +_{c}\nu =\gamma .\supset _{\alpha ,\beta }.\alpha =\beta &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*120·413. \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu -_{c}0=\text{ sm }ʻʻ\mu\)

Dem. \[ \begin{array}{l} \vdash .*119·1 .\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu -_{c}0.&\equiv .\text{N}_{0}\text{c}ʻ\xi +_{c}0=\mu .\exists !\mu .\\ [*110·61.*103·13] &\equiv .\text{Nc}ʻ\xi =\mu .\\ [*103·44·4] &\equiv .\text{N}_{0}\text{c}ʻ\xi =\text{ sm }ʻʻ\mu .\\ [*103·26] &\equiv .\xi \in \text{ sm }ʻʻ\mu \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·414. \[\begin{align}&\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .(\mu -_{c}1)_{\xi }\in \text{N}_{0}\text{C}\\ &[*120·4111.*117·53]\end{align}\]

*120·415. \[\begin{align}&\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0-\iota ʻ1.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .(\mu -_{c}2)_{\xi }\in \text{N}_{0}\text{C}\\ &[*120·4111.*117·551]\end{align}\]

*120·416. \[\begin{align}&\vdash :\nu \in \text{NC induct}.\exists !\gamma -_{c}\nu .\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma \\ &[*120·411.*119·34]\end{align}\]

*120·417. \[\begin{align}&\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0 .\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset .\alpha +_{c}\gamma =(\alpha +_{c}1)+_{c}(\gamma -_{c}1)_{\xi }\\ &[*120·414.*119·35]\end{align}\]

*120·418. \[\begin{align}&\vdash :\nu \in \text{NC induct}.\exists !\text{ sm }_{\xi }ʻʻ\gamma .\gamma \geq \nu .\supset .\alpha +_{c}\gamma =(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )_{\xi }\\ &[*120·4111.*119·35]\end{align}\]

[Pg 218]

*120·42. \(\vdash :\nu \in \text{NC induct}.\exists !\nu .\alpha \neq 0.\supset .\nu \neq \alpha +_{c}\nu \)

Dem. \[ \begin{array}{l} \vdash .*110·61.*120·14.\supset \vdash :\nu\in \text{NC induct}.\supset .\nu =0+_{c}\nu &\qquad \text{(1)}\\ \vdash .*120·41.\supset \vdash :\nu \in \text{NC induct}.\exists !0+_{c}\nu .0+_{c}\nu =\alpha +_{c}\nu .\supset .0=\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash :\nu \in \text{NC induct}.\exists !\nu .\nu =\alpha +_{c}\nu .\supset .\alpha =0:\supset \vdash .\text{Prop} \end{array} \]

*120·422. \(\vdash :\alpha +_{c}1\in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*120·1·124.*91·542.&\supset \vdash :\alpha +_{c}1\in \text{NC induct}.\supset .(\alpha +_{c}1)(+_{c}1)_{\text{po}}0.\\ [*91·52] &\supset .(\exists \beta ).(\alpha +_{c}1)(+_{c}1)\beta .\beta (+_{c}1)_{*}0.\\ [*120·1] &\supset .(\exists \beta ).\alpha +_{c}1=\beta +_{c}1.\beta \in \text{NC induct} &\qquad \text{(1)}\\ \vdash .*120·311. & \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha +_{c}1=\beta +_{c}1.\supset .\alpha =\text{ sm }ʻʻ\beta .\exists !\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*120·15.&\supset \vdash :\text{Hp}.\supset .\alpha \in \text{NC induct} &\qquad \text{(3)}\\ \vdash .(3).*110·4.\supset \vdash .\text{Prop} \end{array} \]

*120·423. \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta }\)

Dem. \[ \begin{array}{l} \vdash .*120·121·124. &\supset \vdash :\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta }.\supset .\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0 &\qquad \text{(1)}\\ \vdash .*120·102.*91·542.&\supset \vdash :\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\supset .\alpha (+_{c}1)_{\text{po}}0_{\eta }.\\ [*91·52] &\supset .(\exists \beta ).\alpha (+_{c}1)\beta .\beta (+_{c}1)_{*}0_{\eta }.\\ [*120·102] &\supset .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta } &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash . \text{Prop} \end{array} \]

*120·4231. \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct}.\supset .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\alpha +_{c}1)_{\eta }=(\beta +_{c} 1)_{\eta }\)

Dem. \[ \begin{array}{l} &\vdash .*10·24.*101·12.*120·12.\supset \\ &\vdash .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(0+_{c}1)_{\eta }=(\beta +_{c} 1)_{\eta } &\qquad \text{(1)}\\ \vdash .*120·121.&\supset \vdash \colon\ldotp \exists !\xi .\supset :\\ &\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\xi +_{c}1)_{\eta }:\\ [*10·23·24] &\supset :(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\\ &(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\gamma +_{c}1)_{\eta } &\qquad \text{(2)}\\ \vdash .*110·4.*13·17.\supset \\ \vdash \colon\ldotp {\sim}\exists !\xi .&\supset :\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .(\xi +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta }:\\ [*10·28] &\supset :(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\\ &(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta } &\qquad \text{(3)}\\ \vdash .(2).(3).&\supset \vdash :(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\\ &(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta } &\qquad \text{(4)}\\ \vdash .(1).(4) \frac{(\xi +_{c}1)_{\eta }}{\xi}.*120·11.\supset \\ &\vdash :\alpha \in \text{N}_{\eta }\text{C induct}.\supset .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\alpha +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta }:\\ \supset \vdash .\text{Prop} \end{array} \]

*120·4232. \[\begin{align}&\vdash :\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\alpha =(\beta +_{c}1)_{\eta }\\ &[*120·423·4231]\end{align}\]

[Pg 219]

*120·424. \(\vdash :\beta \neq 0.\exists !(\alpha +_{c}\beta )_{\xi }.\supset .(\alpha +_{c}\beta )_{\xi }-_{c}1=\alpha +_{c}(\beta -_{c}1)_{\xi }\)

Dem. \[ \begin{array}{l} \vdash .*110·42·62.\supset \vdash :\text{Hp}.\supset .(\alpha +_{c}\beta )_{\xi }\in \text{NC}-\iota ʻ\Lambda -\iota ʻ0.\\ [*120·414.*103·13] \supset .\exists !(\alpha +_{c}\beta )_{\xi }-_{c}1 &\qquad \text{(1)}\\ \vdash .*110·4.*118·21.*120·414.*103·13.\supset \vdash :\text{Hp}.\supset .\exists !(\beta -_{c}1)_{\xi } &\qquad \text{(2)}\\ \vdash .(1).(2).*120·416.\supset \\ \vdash :\text{Hp}. \supset .\{(\alpha +_{c}\beta )_{\xi }-_{c}1\}+_{c}1=\alpha +_{c}\beta .(\beta -_{c}1)_{\xi }+_{c}1=\beta . &\qquad \text{(3)}\\ [*110·56]\supset .\{(\alpha +_{c}\beta )_{\xi }-_{c}1\}+_{c}1=\{\alpha +_{c}(\beta -_{c}1)_{\xi }\}+_{c}1 &\qquad \text{(4)}\\ \vdash .(3).\supset \vdash :\text{Hp}.\supset .\exists ![\{(\alpha +_{c}\beta )_{\xi\}-_{c}1}+_{c}1]_{\xi } &\qquad \text{(5)}\\ \vdash .(4).(5).*120·311.*110·44.\supset \\ \vdash :\text{Hp}.\supset .(\alpha +_{c}\beta )_{\xi }-_{c}1=\alpha +_{c}(\beta -_{c}1)_{\xi }:\supset \vdash .\text{Prop} \end{array} \]

*120·425. \[\begin{align}\vdash \colon\ldotp &(\alpha +_{c}\beta )_{\xi }\in \text{N}_{0}\text{C}-\iota ʻ0.\supset :\\ &(\alpha +_{c}\beta )_{\xi }-_{c}1=\alpha +_{c}(\beta -_{c}1)_{\xi }.\lor.(\alpha +_{c}\beta )_{\xi }-_{c}1=(\alpha -_{c}1)_{\xi }+_{c}\beta \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*110·62.*103·22.\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \neq 0.\lor.\beta \neq 0:\exists !(\alpha +_{c}\beta )_{\xi } &\qquad \text{(1)}\\ \vdash .(1).*120·424.\supset \vdash .\text{Prop} \end{array} \]

*120·426. \(\vdash :\rho \in \text{Cls induct}.\rho \subset \sigma .\exists !\sigma -\rho .\supset .{\sim}(\rho \text{ sm }\sigma ).\text{Nc}ʻ\rho <\text{Nc}ʻ\sigma\)

Dem. \[ \begin{array}{l} \vdash .*110·32.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ\sigma =\text{Nc}ʻ\rho +_{c}\text{Nc}ʻ(\sigma -\rho ) &\qquad \text{(1)}\\ \vdash .*101·14.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ(\sigma -\rho )\neq 0 &\qquad \text{(2)}\\ \vdash .(1).(2).*120·42.*117·222·26.\supset \vdash .\text{Prop} \end{array} \]

*120·427. \[\begin{align}&\vdash :R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !\text{D}ʻR-\text{ᗡ}ʻR.\supset .\text{D}ʻR{\sim}\in \text{Cls induct}\\ &[*120·426.\text{Transp}]\end{align}\]

The above proposition shows that no reflexive class is inductive.

*120·428. \(\vdash :\nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\alpha \neq 0.\supset .\alpha +_{c}\nu >\nu \)

Dem. \[ \begin{array}{l} \vdash .*117·511.*110·4.\supset \vdash :\text{Hp}.&\supset .\alpha >0.\nu \in \text{N}_{0}\text{C} .\\ [*117·561.*110·6] &\supset .\alpha +_{c}\nu \geq \nu &\qquad \text{(1)}\\ \vdash .*120·42.*110·4. &\supset \vdash :\text{Hp}.\supset .\alpha +_{c}\nu \neq \nu &\qquad \text{(2)}\\ \vdash .(1).(2).*117·26.\supset \vdash .\text{Prop} \end{array} \]

*120·429. \(\vdash \colon\ldotp \nu \in \text{NC induct}.\supset :\mu >\nu .\equiv .\mu \geq \nu +_{c}1\)

Dem. \[ \begin{array}{l} \vdash .*120·428. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\mu \in \text{N}_{0}\text{C}.\mu =\nu +_{c}1.\supset .\mu >\nu : &\qquad \text{(1)}\\ [*117·47·12] &\supset :\mu >\nu +_{c}1.\supset .\mu >\nu &\qquad \text{(2)}\\ \vdash .*117·31.&\supset \vdash :\mu >\nu .\supset .(\exists \varpi ).\varpi \in \text{N}_{0}\text{C}.\mu =\nu +_{c}\varpi &\qquad \text{(3)}\\ \vdash .*117·26·12.&\supset \vdash :\mu >\nu .\supset .\mu \neq \nu +_{c}0 &\qquad \text{(4)}\\ \vdash .(3).(4). &\supset \vdash :\mu >\nu .\supset .(\exists \varpi ).\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\mu =\nu +_{c}\varpi .\\ [*117·531] &\supset .(\exists \varpi ).\varpi \geq 1.\mu =\nu +_{c}\varpi .\\ [*117·31] &\supset .(\exists \varpi ,\rho ).\rho \in \text{N}_{0}\text{C}.\varpi =\rho +_{c}1.\mu =\nu +_{c}\varpi .\\ [*13·195] &\supset .(\exists \rho ).\rho \in \text{N}_{0}\text{C}.\mu =\nu +_{c}\rho +_{c}1.\\ [*117·31] &\supset .\mu \geq \nu +_{c}1 &\qquad \text{(5)}\\ \vdash .(1).(2).(5).\supset \vdash .\text{Prop} \end{array} \]

[Pg 220]

The following definition, in which "\(\text{spec}\)" stands for "\(species\)," defines the "species" of a cardinal \(\beta\) as all cardinals which are less than, equal to, or greater than \(\beta\). We cannot prove, unless by assuming the multiplicative axiom, that all cardinals belong to the species of \(\beta\), except in the case where \(\beta\) is an inductive cardinal. In all other cases there may, so far as is known at present, be other cardinals which are neither greater nor less than \(\beta\).

*120·43. \(\text{spec}ʻ\beta =\hat{\alpha} \{\alpha <\beta .\lor.\alpha \geq \beta\} \quad\text{Df}\)

*120·431. \(\vdash \colon\ldotp \alpha \in \text{spec}ʻ\beta .\equiv :\alpha <\beta .\lor.\alpha \geq \beta \quad[(*120·43)]\)

*120·432. \(\vdash \colon\ldotp \alpha \in \text{spec}ʻ\beta .\equiv :\alpha \leq \beta .\lor.\alpha \geq \beta \quad[*117·281.*120·431]\)

*120·433. \[\begin{align}&\vdash \colon\ldotp \text{Nc}ʻ\rho \in \text{spec}ʻ\text{Nc}ʻ\sigma .\equiv :\exists !\text{Cl}ʻ\rho \cap \text{Nc}ʻ\sigma .\lor.\exists !\text{Cl}ʻ\sigma \cap \text{Nc}ʻ\rho \\ &[*117·22.*120·432]\end{align}\]

*120·434. \(\vdash .\text{spec}ʻ\beta \subset \text{N}_{0}\text{C} \quad[*117·105·104·12.*120·432]\)

*120·435. \(\vdash :\beta \in \text{N}_{0}\text{C}.\equiv .\beta \in \text{spec}ʻ\beta .\equiv .\exists !\text{spec}ʻ\beta \quad[*117·104.*120·434]\)

*120·436. \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{spec}ʻ\beta .\equiv :\alpha ,\beta \in \text{N}_{0}\text{C}:(\exists \gamma ):\alpha +_{c}\gamma =\beta .\lor.\beta +_{c}\gamma =\alpha \\ &[*120·432 . *117·31]\end{align}\]

*120·437. \(\vdash :\beta \in \text{N}_{0}\text{C}.\supset .0\in \text{spec}ʻ\beta \quad[*117·5.*120·432]\)

*120·438. \(\vdash :\alpha \in \text{spec}ʻ\beta .\exists !\alpha +_{c}1.\supset .\alpha +_{c}1\in \text{spec}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*120·436.*110·4.&\supset \vdash \colon\ldotp \text{Hp}.\equiv :\alpha ,\beta \in \text{N}_{0}\text{C}.\exists !\alpha +_{c}1:\\ &(\exists \gamma ):\gamma \in \text{N}_{0}\text{C}:\alpha +_{c}\gamma =\beta .\lor.\beta +_{c}\gamma =\alpha &\qquad \text{(1)}\\ \vdash .*110·61. &\supset \vdash :\alpha ,\beta \in \text{N}_{0}\text{C}.\alpha +_{c}0=\beta .\supset .\alpha =\beta .\\ [*13·12·15] &\supset .\alpha +_{c}1=\beta +_{c}1 .\\ [*120·436] &\supset .\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(2)}\\ \vdash .*120·417.&\supset \vdash .\alpha ,\beta ,\gamma \in \text{N}_{0}\text{C}.\gamma \neq 0.\alpha +_{c}\gamma =\beta .\supset .\alpha +_{c}1+_{c}(\gamma -_{c}1)=\beta .\\ [*120·436] &\supset .\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(3)}\\ \vdash .*13·12·15. &\supset \vdash :\alpha ,\beta ,\gamma \in \text{N}_{0}\text{C}.\beta +_{c}\gamma =\alpha .\exists !\alpha +_{c}1.\supset .\beta +_{c}\gamma +_{c}1=\alpha +_{c}1.\\ [*120·436] &\supset .\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(4)}\\ \vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*120·44. \(\vdash :\beta \in \text{N}_{0}\text{C}.\supset .\text{NC induct}-\iota ʻ\Lambda \subset \text{spec}ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*120·437. &\supset \vdash :\text{Hp}.\supset .0\in \text{spec}ʻ\beta &\qquad \text{(1)}\\ \vdash .*120·438.*110·4.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha =\Lambda .\lor.\alpha \in \text{spec}ʻ\beta :\supset :\\ &\alpha +_{c}1=\Lambda .\lor.\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(2)}\\ \vdash .(1).(2).*120.11.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha \in \text{NC induct}.\supset :\\ &\alpha =\Lambda .\lor.\alpha \in \text{spec}ʻ\beta \colon\colon \supset \vdash .\text{Prop} \end{array} \]

*120·441. \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{NC induct}-\iota ʻ\Lambda .\beta \in \text{NC}-\iota ʻ\Lambda .\supset :\alpha <\beta .\lor.\alpha =\text{ sm }ʻʻ\beta .\lor.\alpha >\beta \\ &[*120·44.*103·34]\end{align}\]

[Pg 221]

*120·442. \[\begin{align}\vdash \colon\ldotp \alpha \in \text{NC induct}&-\iota ʻ\Lambda .\beta \in \text{NC}-\iota ʻ\Lambda .\supset :\\ &\alpha < \beta .\equiv .{\sim}(\alpha \geq \beta ):\alpha >\beta .\equiv .{\sim}(\alpha \leq \beta )\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*117·104.*120·441.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha <\beta .\lor.\alpha \geq \beta &\qquad \text{(1)}\\ \vdash .*117·291. &\supset \vdash :\alpha <\beta .\supset .{\sim}(\alpha \geq \beta ) &\qquad \text{(2)}\\ \vdash .(1).(2).*5·17.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha <\beta .\equiv .{\sim}(\alpha \geq \beta ) &\qquad \text{(3)}\\ \text{Similarly}\quad &\vdash \colon\ldotp \text{Hp}.\supset :\alpha >\beta .\equiv .{\sim}(\alpha \leq \beta ) &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

*120·45. \(\vdash :\alpha ,\beta \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}\)

Dem. \[ \begin{array}{l} \vdash .*110·6.\supset \vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}0_{\xi })_{\xi }\in \text{N}_{\xi }\text{C induct} &\qquad \text{(1)}\\ \vdash .*120·121.*118·25.\supset\\ \vdash :(\alpha +_{c}\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}.\supset .\{\alpha +_{c}(\beta +_{c}1)_{\xi }\}_{\xi }\in \text{N}_{\xi }\text{C induct} &\qquad \text{(2)}\\ \vdash .(1).(2).*120·11.\supset \vdash .\text{Prop} \end{array} \]

*120·4501. \(\vdash :\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha +_{c}\beta \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·15.\supset \vdash :\text{Hp}.&\supset .\text{ sm }_{\xi }ʻʻ\alpha ,\text{ sm }_{\xi }ʻʻ\beta \in \text{N}_{\xi }\text{C induct}.\\ [*120·45] &\supset .(\text{ sm }_{\xi }ʻʻ\alpha +_{c}\text{ sm }_{\xi }ʻʻ\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}.\\ [*118·23] &\supset .(\alpha +_{c}\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}:\supset \vdash .\text{Prop} \end{array} \]

The following proposition is a lemma in the proof of *120·452.

*120·451. \[\begin{align}&\vdash \colon\ldotp \gamma =(\alpha +_{c}\beta )_{\xi }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda :\\ &\exists !(\gamma +_{c}1)_{\xi }.(\gamma +_{c}1)_{\xi }=(\alpha ʻ+_{c}\beta ʻ)_{\xi }:\supset .\alpha ʻ,\beta ʻ\in \text{NC induct}-\iota ʻ\Lambda\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*120·414·124.*110·42.&\supset \vdash :\exists !(\gamma +_{c}1)_{\xi }.\supset .\{(\gamma +_{c}1)_{\xi }-_{c}1\}_{\xi}\in \text{N}_{0}\text{C}.\\ [*119·32] &\supset .\gamma =\{(\gamma +_{c}1)_{\xi}-_{c}1\}_{\xi } &\qquad \text{(1)}\\ \vdash .(1).*120·124.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma =\{(\alpha'+ \beta')_{\xi}-_{c}1\}_{\xi }.(\alpha'+ \beta')_{\xi }\neq 0.\exists !(\alpha'+ \beta')_{\xi }:\\ [*120·425]&\supset :\gamma =\{\alpha'+_{c}(\beta'-_{c}1)_{\xi }\}_{\xi }.\lor.\gamma =\{(\alpha'-_{c}1)_{\xi }+_{c}\beta'\}_{\xi }:\\ [\text{Hp}] &\supset :\alpha',(\beta'-_{c}1)_{\xi }\in \text{NC induct}-\iota ʻ\Lambda .\lor.(\alpha'-_{c}1)_{\xi },\beta'\in \text{NC induct}-\iota ʻ\Lambda :\\ [*119·11] &\supset :\alpha',\beta'\in \text{NC induct}-\iota ʻ\Lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

This proposition could be extended to greater generality as regards types; but its sole use is as a lemma.

*120·452. \(\vdash :\alpha +_{c}\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \)

Dem. \[ \begin{array}{l} \vdash .*110·4.\text{Transp}.&\supset \vdash :\gamma =\Lambda .\supset .(\gamma +_{c}1)_{\eta }=\Lambda &\qquad \text{(1)}\\ \vdash .*120·451.&\supset \vdash \colon\colon \gamma =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda :\supset \colon\ldotp \\ &(\gamma +_{c}1)_{\eta }=\Lambda :\lor:(\gamma +_{c}1)_{\eta }=(\alpha ʻ+_{c}\beta ʻ)_{\eta }.\supset _{\alpha ʻ,\beta ʻ}.\alpha ʻ,\beta ʻ\in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(2)}\\ \vdash .(1).(2). &\supset \vdash \colon\colon \gamma =\Lambda :\lor:\gamma =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \colon\ldotp \supset \colon\ldotp\\ &(\gamma +_{c}1)_{\eta }=\Lambda :\lor:(\gamma +_{c}1)_{\eta }=(\alpha ʻ+_{c}\beta ʻ)_\eta .\supset _{\alpha ʻ,\beta ʻ}.\alpha ʻ,\beta ʻ\in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(3)}\\ \vdash .*110·62.*120·12.&\supset \vdash :0=(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(4)}\\ \vdash .(3).(4).*120·11.&\supset \vdash \colon\colon \gamma \in \text{N}_{\eta }\text{C induct}.\supset \colon\ldotp \\ &\gamma =\Lambda :\lor:\gamma =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \colon\colon \\ [*13·15] &\supset \vdash \colon\ldotp (\alpha +_{c}\beta )_{\eta }\in \text{N}_{\eta }\text{C induct}.\supset :\\ &(\alpha +_{c}\beta )_{\eta }=\Lambda .\lor.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

[Pg 222]

In the last line but one of the above proof, we substitute for the \(\phi \xi\) of *120·11 the function \[ \xi =\Lambda :\lor:\xi =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda . \]

The following propositions are chiefly required as leading to *120·4621·4622·47, which are useful in proving propositions concerning all inductive cardinals other than zero.

*120·46. \(\vdash :\alpha \in \text{NC}.\gamma \in \text{N}_{\eta }\text{C induct}.\supset .(\alpha +_{c}\gamma )_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha \)

Dem. \[ \begin{array}{l} \vdash .*110·6.*118·241.&\supset \vdash :\alpha \in \text{NC}.\supset .(\alpha +_{c}0)_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha &\qquad \text{(1)}\\ \vdash .*90·172.*118·25.&\supset \vdash :(\alpha +_{c}\gamma )_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .\\ &\{\alpha +_{c}(\gamma +_{c}1)_{\eta }\}_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*120·11.&\supset \vdash .\text{Prop} \end{array} \]

*120·461. \(\vdash :\alpha \in \text{NC}.\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma )_{\eta }\)

Dem. \[ \begin{array}{l} \vdash .*110·6.*118·23. &\supset \vdash :\alpha \in \text{NC}.\beta =\text{ sm }_{\eta }ʻʻ\alpha .\supset .\beta =(a+_{c}0)_{\eta } &\qquad \text{(1)}\\ \vdash .*120·121.*118·25.&\supset \vdash :\beta =(\alpha +_{c}\gamma )_{\eta }.\gamma \in \text{N}_{\eta }\text{C induct}.\supset .\\ &(\beta +_{c}1)_{\eta }=\{\alpha +_{c}(\gamma +_{c}1)_{\eta }\}_{\eta }.(\gamma +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct} &\qquad \text{(2)}\\ \vdash .(1).(2).*90·112.\supset \vdash .\text{Prop} \end{array} \]

*120·462. \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{NC}.\supset :(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma )_{\eta }.\equiv .\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha \\ &[*120·46·461]\end{align}\]

*120·4621. \(\vdash \colon\ldotp \alpha \in \text{NC}.\exists !\beta .\supset :\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha \)

Dem. \[ \begin{array}{l} \vdash .*120·461.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ \beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .&\supset .(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma )_{\eta }.\\ [*110·4] &\supset .(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\beta =(\alpha +_{c}\gamma )_{\eta }.\\ [*120·15.*118·201] &\supset .(\exists \gamma ).\text{ sm }_{\xi }ʻʻ\gamma \in \text{N}_{\xi }\text{C induct}.\text{ sm }_{\xi }ʻʻ\beta =(\alpha +_{c}\gamma )_{\xi }.\\ [*118·24.*120·14] &\supset .(\exists \gamma').\gamma'\in \text{N}_{\xi }\text{C induct}.\text{ sm }_{\xi }ʻʻ\beta =(\alpha +_{c}\gamma')_{\xi }.\\ [*120·462]& \supset .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·4622. \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{NC}.\beta \in \text{NC}(\eta ).\exists !\text{ sm }_{\xi }ʻʻ\beta .\supset :\\ &\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\equiv .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*110·4.*37·29.*120·461.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha .\supset .\exists !\text{ sm }_{\xi }ʻʻ\alpha .\exists !\alpha . &\qquad \text{(1)}\\ [*100·52] &\supset .\text{ sm }_{\xi }ʻʻ\alpha \in \text{NC} &\qquad \text{(2)}\\ \vdash .*120·4621.(2).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ &\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha .\supset .\text{ sm }_{\eta }ʻʻ\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\text{ sm }_{\xi }ʻʻ\alpha .\\ [*102·87.\text{Hp}.(1)] &\supset .\text{ sm }_{\eta }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\\ [*103·34] &\supset .\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha &\qquad \text{(3)}\\ \vdash .*37·29.*120·4621.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

[Pg 223]

It is on this proposition that the irrelevance of types in the consideration of inductive cardinals depends.

*120·463. \[\begin{align}\vdash \colon\colon .\alpha \in \text{NC}.\supset \colon\colon &(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma )_{\eta }.\equiv \colon\ldotp \\ &\xi \in \mu .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \mu :\text{ sm }_{\eta }ʻʻ\alpha \in \mu :\supset _{\mu }.\beta \in \mu \\ [*120·462.*90·11]\end{align}\]

*120·47. \[\begin{align}&\vdash \colon\colon \beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \mu :1_{\eta }\in \mu :\supset _{\mu }.\beta \in \mu \\ &[*120·423·463]\end{align}\]

Thus mathematical induction starting from 1 will apply to all inductive cardinals except 0. Similar propositions can be similarly proved for 2, 3, ....

*120·471. \(\vdash :(\exists \alpha ).\alpha \in \text{NC induct}-\iota ʻ0.f\alpha .\equiv .(\exists \beta ).\beta \in \text{NC induct}.f(\beta +_{c}1)\)

Dem. \[ \begin{array}{l} \vdash .*120·423.\supset \\ \vdash :(\exists \alpha ).\alpha \in \text{NC induct}-\iota ʻ0.f\alpha .&\equiv .(\exists \beta ).\beta \in \text{NC induct}.\alpha =\beta +_{c}1.f\alpha .\\ [*13·195] &\equiv .(\exists \beta ).\beta \in \text{NC induct}.f(\beta +_{c}1):\supset \vdash .\text{Prop} \end{array} \]

*120·472. \[\begin{align}\vdash :&(\exists \alpha ).\alpha \in \text{NC induct}-\iota ʻ0-\iota ʻ1.f\alpha .\equiv .\\ &(\exists \beta ).\beta \in \text{NC induct}-\iota ʻ0.f(\beta +_{c}1).\equiv .(\exists \gamma ).\gamma \in \text{NC induct}.f(\gamma +_{c}2)\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*120·471.\supset \\ \vdash :(\exists \alpha ).\alpha \in &\text{NC induct}-\iota ʻ0-\iota ʻ1.f\alpha .\equiv .\\ &(\exists \beta ).\beta \in \text{NC induct}.\beta +_{c}1\neq 1.f(\beta +_{c}1).\\ [*120·42.*110·641] &\equiv .(\exists \beta ).\beta \in \text{NC induct}-\iota ʻ0.f(\beta +_{c}1). &\qquad \text{(1)}\\ [*120·471] &\equiv .(\exists \gamma ).\gamma \in \text{NC induct}.f(\gamma +_{c}1+_{c}1).\\ [*110·643] &\equiv .(\exists \gamma ).\gamma \in \text{NC induct}.f(\gamma +_{c}2) &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*120·473. \[\begin{align}\vdash \colon\ldotp \phi 1:\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi \xi .&\supset _{\xi }.\phi (\xi +_{c}1):\supset :\\ &\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\supset .\phi \xi\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*120·122.*101·22.&\supset \vdash :\phi 1.\supset .1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi 1 &\qquad \text{(1)}\\ \vdash .*120·121·124.&\supset \vdash :\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\supset .\xi +_{c}1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0 &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi 1:\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi \xi .\supset _{\xi }.\\ &\xi +_{c}1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi (\xi +_{c}1) &\qquad \text{(3)}\\ \vdash .(3).*120·47 \frac{\hat{\xi} (\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi \xi )}{\mu}.&\supset \vdash .\text{Prop} \end{array} \]

[Pg 224]

*120·48. \[\begin{align}&\vdash :\beta \in \text{NC induct}.\beta \geq \alpha .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda \\ &[*120·452.*117·31]\end{align}\]

Thus every cardinal which is not greater than every inductive cardinal is an inductive cardinal.

*120·481. \(\vdash :\eta \in \text{Cls induct}.\xi \subset \eta .\supset .\xi \in \text{Cls induct} \quad[*117·222.*120·21·48]\)

Thus if any inductive class can be found which contains a given class, the given class is also inductive.

*120·49. \(\vdash :\alpha \in \text{NC}-\text{NC induct}-\iota ʻ\Lambda .\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha >\beta\)

Dem. \[ \begin{array}{l} \vdash .*120·48.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .{\sim}(\beta \geq \alpha ) &\qquad \text{(1)}\\ \vdash .*120·441. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha >\beta .\lor.\beta \geq \alpha &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

Thus every non-inductive cardinal (except \(\Lambda\)) is greater than every inductive cardinal (except \(\Lambda\)).

*120·491. \(\vdash \colon\ldotp \xi {\sim}\in \text{Cls induct}.\equiv :\beta \in \text{NC induct}.\supset _{\beta }.\exists !\beta \cap \text{Cl}ʻ\xi\)

Dem. \[ \begin{array}{l} \vdash .*120·49.&\supset \vdash :\xi {\sim}\in \text{Cls induct}.\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\text{N}_{0}\text{c}ʻ\xi >\beta .\\ [*120·429.*117·12]&\supset .\text{N}_{0}\text{c}ʻ\xi \geq \beta +_{c}1.\exists !\beta \cap \text{Cl}ʻ\xi &\qquad \text{(1)}\\ \vdash .(1).*117·104·12.*103·13.&\supset \vdash :\xi {\sim}\in \text{Cls induct}.\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\beta +_{c}1\neq \Lambda &\qquad \text{(2)}\\ \vdash .(2).*101·12.*102·13.\supset \\ \vdash \colon\ldotp \xi {\sim}\in \text{Cls induct}.&\supset :\beta \in \text{NC induct}.\supset .\beta \neq \Lambda &\qquad \text{(3)}\\ \vdash .(1).(3).&\supset \vdash :\xi {\sim}\in \text{Cls induct}.\beta \in \text{NC induct}.\supset .\exists !\beta \cap \text{Cl}ʻ\xi &\qquad \text{(4)}\\ \vdash .*120·121.\supset \\ \vdash \colon\ldotp \beta \in \text{NC induct}.&\supset _{\beta }.\exists !\beta \cap \text{Cl}ʻ\xi :\supset :\beta \in \text{NC induct}.\supset _{\beta }.\exists !(\beta +_{c}1)\cap \text{Cl}ʻ\xi .\\ [*117·242.*120·429] &\supset _{\beta }.\text{Nc}ʻ\xi > \beta .\\ [*117·42.(*117·03)] &\supset _{\beta }.\text{N}_{0}\text{c}ʻ\xi \neq \beta :\\ [*13·196] &\supset :\text{N}_{0}\text{c}ʻ\xi {\sim}\in \text{NC induct}:\\ [*120·21] &\supset :\xi {\sim}\in \text{Cls induct} &\qquad \text{(5)}\\ \vdash .(4).(5).\supset \vdash .\text{Prop} \end{array} \]

*120·492. \[\begin{align}&\vdash :\alpha \in \text{NC}-\text{NC induct}.\beta \geq \alpha .\supset .\beta \in \text{NC}-\text{NC induct}\\ &[*120·48.\text{Transp}]\end{align}\]

In virtue of *120·491, a class \(\xi\) which is not inductive contains sub-classes having 0, 1, 2, 3, ... terms. If we take the successive classes of sub-classes \[ 0\cap \text{Cl}ʻ\xi , 1\cap \text{Cl}ʻ\xi , 2\cap \text{Cl}ʻ\xi , ..., \] these are mutually exclusive, and all exist provided \(\Lambda\) is not an inductive[Pg 225] cardinal, i.e. provided the axiom of infinity holds. Thus if the axiom of infinity holds, we get \(\aleph_{0}\) classes of sub-classes contained in any non-inductive class. It follows, as we shall see later, that if \(\xi\) is a non-inductive class, \(\text{Cl}ʻ\text{Cl}ʻ\xi\) is a reflexive class. This seems to be the nearest approach possible to identifying the two definitions of finite and infinite when the multiplicative axiom is not assumed. When the multiplicative axiom is assumed as well as the axiom of infinity, we pick out one class from \(1\cap \text{Cl}ʻ\xi\), one from \(2\cap\text{Cl}ʻ\xi\), and so on; then, forming the logical sum of all these classes, we get \(\aleph _{0}\) terms which are members of \(\xi\). Hence it follows that \(\xi\) is a reflexive class; for, as we shall see later, a reflexive class is one which contains sub-classes of \(\aleph _{0}\) terms. Thus with the help of the multiplicative axiom, the two definitions of finite and infinite can be identified.

*120·493. \[\begin{align}&\vdash \colon\ldotp \sigma \in \text{Cls induct}.\supset :\\ &\text{Nc}ʻ\xi <\text{Nc}ʻ\sigma .\equiv .(\exists \rho ).\rho \text{ sm }\xi .\rho \subset \sigma .\exists !\sigma -\rho .\equiv .\exists !\text{Nc}ʻ\xi \cap \text{Cl}ʻ\sigma -\iota ʻ\sigma\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*117·26·221.&\supset \vdash \colon\ldotp \text{Nc}ʻ\xi <\text{Nc}ʻ\sigma .\supset :{\sim}(\xi \text{ sm }\sigma ):(\exists \rho ).\rho \text{ sm }\xi .\rho \subset \sigma :\\ [*73·3·37] &\supset :(\exists \rho ).\rho \text{ sm }\xi .\rho \subset \sigma .\rho \neq \sigma &\qquad \text{(1)}\\ \vdash .*120·481.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\rho \subset \sigma .\exists !\sigma -\rho .\supset .\rho \in \text{Cls} \text{induct}.\rho \subset \sigma .\exists !\sigma -\rho .\\ [*120·426] &\supset .\text{Nc}ʻ\rho <\text{Nc}ʻ\sigma :\\ [*100*321] &\supset :\rho \text{ sm }\xi .\rho \subset \sigma .\exists !\sigma -\rho .\supset .\text{Nc}ʻ\xi <\text{Nc}ʻ\sigma &\qquad \text{(2)}\\ \vdash .(1).(2).*24·6.\supset \vdash .\text{Prop} \end{array} \]

*120·5. \(\vdash :\alpha ,\beta \in \text{NC induct}.\exists !\alpha \times _{c}\beta .\supset .\alpha \times _{c}\beta \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*113·203.&\supset \vdash :\alpha \in \text{NC induct}.\exists !\alpha \times _{c}0.\supset .\alpha \in \text{NC}-\iota ʻ\Lambda .\\ [*113·601] &\supset .\alpha \times _{c}0=0.\\ [*120·12] &\supset .\alpha \times _{c}0\in \text{NC induct} &\qquad \text{(1)}\\ \vdash .*113·671. &\supset \vdash .\alpha \times _{c}(\beta +_{c}1)=(\alpha \times _{c}\beta )+_{c}\alpha .\\ [*120·4501.*113·203] &\supset \vdash :\alpha \in \text{NC induct}.\alpha \times _{c}\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\\ &\alpha \times _{c}(\beta +_{c}1)\in \text{NC induct} &\qquad \text{(2)}\\ \vdash .(1).(2).*120·13.\supset \vdash .\text{Prop} \end{array} \]

The restriction involved in \(\exists !\alpha \times _{c}\beta\) in the hypothesis of the above proposition is not necessary if we assume that the axiom of infinity must fail in any one type if it fails in any other, i.e. \[ \Lambda \cap tʻ\alpha \in \text{NC induct}.\supset .\Lambda \cap tʻ\beta \in \text{NC induct}, \] [Pg 226]where \(\alpha\) and \(\beta\) are any two objects of any two types. To prove this proposition would require assumptions, as to the interrelation of various types, which have not been made in our previous proofs.

*120·51. \(\vdash :\alpha ,\beta ,\gamma ,\in \text{NC induct}.\alpha \neq 0.\exists !\alpha \times _{c}\beta .\alpha\times _{c}\beta =\alpha \times _{c}\gamma .\supset .\beta =\text{ sm }ʻʻ\gamma\)

This proposition establishes the uniqueness of division among inductive cardinals.

Dem. \[ \begin{array}{l} \vdash . *120·44·436. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists \delta ):\beta =\gamma +_{c}\delta .\lor.\gamma =\beta +_{c}\delta &\qquad \text{(1)}\\ \vdash . *113·43. &\supset \vdash :\text{Hp}.\beta =\gamma +_{c}\delta .\supset .\alpha \times _{c}\gamma =(\alpha \times _{c}\gamma )+_{c}(\alpha \times _{c}\delta ).\\ [*120·42.\text{Transp}] &\supset .\alpha \times _{c}\delta =0.\\ [*113·602] &\supset .\delta =0.\\ [*110·6] &\supset .\beta = \text{ sm }ʻʻ\gamma &\qquad \text{(2)}\\ \text{Similarly}\quad &\vdash :\text{Hp}.\gamma =\beta +_{c}\delta .\supset .\gamma =\text{ sm }ʻʻ\beta .\\ [*100·53.*113·203] &\supset .\beta =\text{ sm }ʻʻ\gamma &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

If \(\beta\), \(\gamma\) in the above are typically ambiguous symbols, such as \[ 0, 1, 2, ... \text{Nc}ʻ\rho , \text{Nc}ʻ\sigma , ..., \] we have \(\beta = \gamma\); for in this case, \(\beta =\text{ sm }ʻʻ\beta.\gamma =\text{ sm }ʻʻ\gamma\). Also if \(\beta\) and \(\gamma\) are of the same type, we have \(\beta=\gamma\), in virtue of *103·43. Hence "\(\beta =\gamma\)" may, with truth, be substituted for "\(\beta =\text{ sm }ʻʻ\gamma\)" in the above proposition, since the result is true whenever significant. But in this form the proposition gives less information, since it tells us nothing as to what happens when \(\beta\) and \(\gamma\) are not of the same type.

*120·511. \(\vdash :\alpha ,\beta \in \text{NC induct}.\alpha \neq 0.\exists !\alpha .\alpha \times _{c}\beta =\alpha .\supset .\beta =1\)

Dem. \[ \begin{array}{l} \vdash .*113·621. \supset \vdash :\text{Hp}.\supset .\alpha \times _{c}\beta =\alpha \times _{c} 1 &\qquad \text{(1)}\\ \vdash .(1).*120·51.*101·28.\supset \vdash .\text{Prop} \end{array} \]

*120·512. \(\vdash :\alpha \times _{c}\beta \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda .\supset .\alpha ,\beta \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda \)

Dem. \[ \begin{array}{l} \vdash . *113·602·203.\supset \vdash :\text{Hp}.&\supset .\alpha ,\beta \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda &\qquad \text{(1)}\\ \vdash .(1).*117·62.\supset \vdash :\text{Hp}.&\supset .\alpha \times _{c}\beta \geq \alpha .\alpha \times _{c}\beta \geq \beta .\\ [*120·48] &\supset .\alpha ,\beta \in \text{NC induct} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*120·513. \(\vdash :\alpha \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda .\alpha \times _{c}\beta =\alpha .\supset .\beta =1 \quad[*120·511·512]\)

This proposition does not hold when \(\alpha\) is a non-inductive cardinal.

*120·52. \(\vdash :\alpha ,\beta \in \text{NC induct}.\exists !\alpha ^\beta .\supset .\alpha ^\beta \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*116·203·301.&\supset \vdash :\alpha \in \text{NC induct}.\exists !\alpha ^{0}.\supset .\alpha ^{0}=1.\\ [*120·122] &\supset .\alpha ^{0}\in \text{NC induct} &\qquad \text{(1)}\\ \vdash . *116·321·52. &\supset \vdash :\exists !\alpha ^{\beta +_{c}1}.\supset .\alpha ^{\beta +_{c}1}=\alpha ^\beta \times _{c}\alpha .\\ [*120·5] &\supset \vdash :\alpha \in \text{NC induct}.\alpha ^\beta \in \text{NC induct}.\exists !\alpha ^{\beta +_{c}1}.\supset .\\ &\alpha ^{\beta +_{c}1}\in \text{NC induct} &\qquad \text{(2)}\\ \vdash .*116·52.*113·204.&\supset \vdash :\alpha ^\beta =\Lambda .\supset .\alpha ^{\beta +_{c}1}=\Lambda &\qquad \text{(3)}\\ \vdash .(1).(2).(3).*120·11.\supset \vdash .\text{Prop} \end{array} \]

[Pg 227]

*120·53. \(\vdash :\alpha ,\beta ,\gamma \in \text{NC induct}.\alpha \neq 0.a\neq 1.\exists !\alpha ^\beta .\alpha ^\beta =\alpha ^\gamma .\supset .\beta =\text{ sm }ʻʻ\gamma \)

Dem. \[ \begin{array}{l} \vdash .*116·203. &\supset \vdash :\exists !\alpha ^\beta .\supset .\exists !\beta &\qquad \text{(1)}\\ \vdash .*120·44·436. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists \delta ):\beta =\gamma +_{c}\delta .\lor.\gamma =\beta +_{c}\delta &\qquad \text{(2)}\\ \vdash .*118·01.*116·52. &\supset \vdash :\beta =\gamma +_{c}\delta .\exists !\beta .\supset .\alpha ^\beta =\alpha ^\gamma \times _{c}\alpha ^\delta :\\ [*13·171.*118·01.(1)] & \supset \vdash :\alpha ^\beta =\alpha ^\gamma .\beta =\gamma +_{c}\delta .\exists !\alpha ^\beta .\supset .\alpha ^\gamma =\alpha ^\gamma \times _{c}\alpha ^\delta &\qquad \text{(3)}\\ \vdash .*120·52.*116·35.(1).&\supset :\text{Hp}.\beta =\gamma +_{c}\delta .\supset .a^\gamma \in \text{NC induct}-\iota ʻ\Lambda -\iota ʻ0.\exists !\beta .\\ [(3).*120·513] & \supset .\alpha ^\delta =1.\\ [*117·592] &\supset .\delta =0.\\ [*110·6] &\supset .\beta =\text{ sm }ʻʻ\gamma &\qquad \text{(4)}\\ \text{Similarly}\quad &\vdash :\text{Hp}.\gamma =\beta +_{c}\delta .\supset .\gamma =\text{ sm }ʻʻ\beta .\\ [*100·53.(1)] &\supset .\beta =\text{ sm }ʻʻ\gamma &\qquad \text{(5)}\\ \vdash .(2).(4).(5).\supset \vdash .\text{Prop} \end{array} \]

If \(\alpha\), \(\beta\), \(\gamma\) are typically ambiguous symbols, we have \(\beta =\gamma\) in the conclusion of the above proposition, instead of \(\beta =\text{ sm }ʻʻ\gamma\). Also if \(\beta\) and \(\gamma\) are of the same type, \(\beta =\gamma\); thus \(\beta=\gamma\) whenever "\(\beta =\gamma\)" is significant.

*120·54. \(\vdash :\xi ,\rho \in \text{Cls induct}.\exists !\xi .\rho \subset \sigma .\exists !\sigma - \rho .\supset .(\text{Nc}ʻ\rho )^{\text{Nc}ʻ\xi }<(\text{Nc}ʻ\sigma )^{\text{Nc}ʻ\xi }\)

For the proof, which is here given shortly, compare *117·58.

Dem. \[ \begin{array}{l} \vdash .*35·432·82.*80·15.*116·12. &\supset \vdash :\text{Hp}.\supset .(\rho \uparrow \xi )_{\Delta }ʻ\xi \subset (\sigma \uparrow \xi )_{\Delta }ʻ\xi .\\ &\exists !(\sigma \uparrow \xi )_{\Delta }ʻ\xi -(\rho \uparrow \xi )_{\Delta }ʻ\xi &\qquad \text{(1)}\\ \vdash .*120·52.*116·15·251.*120·2.&\supset \vdash :\text{Hp}.\supset .(\rho \uparrow \xi )_{\Delta }ʻ\xi \in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).(2).*120·426.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ(\rho \uparrow \xi )_{\Delta }ʻ\xi < \text{Nc}ʻ(\sigma \uparrow \xi )_{\Delta }ʻ\xi :\supset \vdash .\text{Prop} \end{array} \]

*120·541. \(\vdash :\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda .\alpha \neq 0.\beta <\gamma .\supset .\beta ^\alpha <\gamma ^\alpha \quad[*120·54·493]\)

*120·542. \(\vdash :\alpha ,\gamma \in \text{NC induct}-\iota ʻ\Lambda .\alpha \neq 0.\beta >\gamma .\supset .\beta ^\alpha >\gamma ^\alpha \quad[*120·541]\)

*120·55. \(\vdash :\alpha ,\beta ,\gamma \in \text{NC induct}.\alpha \neq 0.\exists !\beta ^\alpha .\beta ^\alpha =\gamma ^\alpha .\supset .\beta =\text{ sm }ʻʻ\gamma\)

Dem. \[ \begin{array}{l} \vdash .*120·541·542.\supset \vdash :\text{Hp}.&\supset .{\sim}(\beta <\gamma ).{\sim}(\beta >\gamma ).\\ [*120·441] &\supset .\beta =\text{ sm }ʻʻ\gamma :\supset \vdash .\text{Prop} \end{array} \]

*120·56. \(\vdash :\alpha \geq 2.\alpha ^\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\beta \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*117·581.\supset \vdash :\text{Hp}.&\supset .\alpha ^\beta \geq 2^\beta .\\ [*117·661] & \supset .\alpha ^\beta >\beta &\qquad \text{(1)}\\ \vdash .(1).*120·48.\supset \vdash .\text{Prop} \end{array} \]

*120·561. \(\vdash :\beta \geq 1.\alpha ^\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*117·591.*116·321.\supset \vdash :\text{Hp}.\supset .\alpha ^\beta \geq \alpha &\qquad \text{(1)}\\ \vdash .(1).*120·48.\supset \vdash .\text{Prop} \end{array} \]

[Pg 228]

*120·57. \(\vdash :\mu \in \text{NC induct}-\iota ʻ\Lambda .\supset .\text{Nc}ʻ\hat{\nu} (\nu \leq \mu )=\mu +_{c}1\)

Here "\(\mu +_{c}1\)" is necessarily in a higher type than "\(\mu\)," because it applies to a class of which \(\mu\) is a member.

Dem. \[ \begin{array}{l} \vdash .*117·511.&\supset \vdash .\text{Nc}ʻ\hat{\nu} (\nu \leq 0)\in 1 &\qquad \text{(1)}\\ \vdash . *110·4. &\supset \vdash :\mu =\Lambda .\supset .\mu +_{c}1=\Lambda &\qquad \text{(2)}\\ \vdash .*120·429·442.\supset \\ &\vdash :\mu \in \text{NC induct}.\exists !\mu +_{c}1.\supset .\hat{\nu} (\nu \leq \mu )=\hat{\nu} (\nu <\mu +_{c}1).\\ [*117·104·105] &\supset .\hat{\nu} (\nu \leq \mu +_{c}1)=\hat{\nu} (\nu \leq \mu )\cup \iota ʻ(\mu +_{c}1) &\qquad \text{(3)}\\ \vdash .*120·428.&\supset \vdash :\text{Hp}(3).\supset .\mu +_{c}1{\sim}\in \hat{\nu} (\nu \leq \mu ) &\qquad \text{(4)}\\ \vdash .(3).(4).*110·631.\supset \\ &\vdash :\text{Hp}(3).\text{Nc}ʻ\hat{\nu} (\nu \leq \mu )=\mu +_{c}1.\supset .\text{Nc}ʻ\hat{\nu} (\nu \leq \mu +_{c}1)=\mu +_{c}2 &\qquad \text{(5)}\\ \vdash .(2).(5).&\supset \vdash \colon\ldotp \mu \in \text{NC induct}:\mu =\Lambda .\lor.\text{Nc}ʻ\hat{\nu} (\nu \leq \mu )=\mu +_{c}1:\supset :\\ &\mu +_{c}1=\Lambda .\lor.\text{Nc}ʻ\hat{\nu} (\nu \leq \mu +_{c}1)=\mu +_{c}2 &\qquad \text{(6)}\\ \vdash .(1).(6).*120·13.\supset \vdash .\text{Prop} \end{array} \]

*120·6. \(\vdash :(\exists \gamma ).\gamma >\alpha .\gamma \subset tʻ\eta .\supset .\exists !(\alpha +_{c}1)\cap tʻ\eta \)

Dem. \[ \begin{array}{l} \vdash .*117·1.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \gamma ,\rho ,\sigma ).\text{N}_{0}\text{c}ʻ\rho =\alpha .\text{N}_{0}\text{c}ʻ\sigma =\gamma .\exists !\text{Nc}ʻ\rho \cap \text{Cl}ʻ\sigma .{\sim}\exists !\text{Nc}ʻ\sigma \cap \text{Cl}ʻ\rho :\\ [*100·1]&\supset :(\exists \gamma ,\rho ,\sigma ,\xi ).\text{N}_{0}\text{c}ʻp=\alpha .\text{N}_{0}\text{c}ʻ\sigma =\gamma .\xi \text{ sm }\rho .\xi \subset \sigma .\xi \neq \sigma :\\ [*24·6] &\supset :(\exists \gamma ,\rho ,\sigma ,\xi ,x).\text{N}_{0}\text{c}ʻp=\alpha .\text{N}_{0}\text{c}ʻ\sigma =\gamma .\xi \text{ sm }\rho .x\in \sigma -\xi :\\ [*110·631]&\supset :(\exists \xi ,x).\xi \cup \iota ʻx\in \alpha +_{c}1\cap tʻ\eta \colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·61. \(\vdash :\exists !\text{N}_{0}\text{C}\cap t^{3}ʻx-\text{NC induct}.\supset .\text{Infin ax}(x)\)

Dem. \[ \begin{array}{l} \vdash .*120·49.&\supset \vdash \colon\ldotp \gamma \in \text{N}_{0}\text{C}\cap t^{3}ʻx-\text{NC induct}.\supset :\\ &\alpha \in \text{NC induct}.\exists !\alpha . \supset _{\alpha }.\gamma >\alpha .\gamma \subset t^{2}ʻx.\\ [*120·6] &\supset _{\alpha }.\exists !\alpha +_{c}1\cap t^{2}ʻx &\qquad \text{(1)}\\ \vdash .(1).*101·12.*120·13.\supset \\ \vdash \colon\ldotp \gamma \in \text{N}_{0}\text{C}-\text{NC induct}.&\supset :\alpha \in \text{NC induct}. \supset _{\alpha }.\exists !\alpha (x)\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·611. \(\vdash :\beta \in \text{Cls induct}.\beta \subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ\beta\)

Dem. \[ \begin{array}{l} \vdash .*80·26.&\supset \vdash .\exists !P_{\Delta }ʻ\Lambda .\\ [\text{Simp}] &\supset \vdash :\Lambda \subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ\Lambda &\qquad \text{(1)}\\ \vdash .*80·94.&\supset \vdash :\exists !P_{\Delta }ʻ\beta .z\in \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ(\beta \cup \iota ʻz):\\ [\text{Syll}] &\supset \vdash \colon\ldotp \beta \subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ\beta :\supset :\beta \subset \text{ᗡ}ʻP.z\in \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ(\beta \cup \iota ʻz):\\ [*51·238] &\supset :\beta \cup \iota ʻz\subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ(\beta \cup \iota ʻz) &\qquad \text{(2)}\\ \vdash .(1).(2).*120·26.\supset \vdash .\text{Prop} \end{array} \]

[Pg 229]

*120·62. \(\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .\supset .\exists !{\in}_{\Delta}ʻ\kappa \)

Dem. \[ \begin{array}{l} \vdash .*83·9. &\supset \vdash .\exists !{\in}_{\Delta}ʻ\Lambda &\qquad \text{(1)}\\ \vdash . *83·904. &\supset \vdash :\exists !{\in}_{\Delta}ʻ\kappa .\exists !\alpha .\supset .\exists !{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ):\\ [\text{Syll}] &\supset \vdash \colon\ldotp \Lambda {\sim}\in \kappa .\supset .\exists !{\in}_{\Delta}ʻ\kappa :\supset :\Lambda {\sim}\in \kappa .\exists !\alpha .\supset .\exists !{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ):\\ [*24·54] &\supset :\Lambda {\sim}\in (\kappa \cup \iota ʻ\alpha ).\supset .\exists !{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ) &\qquad \text{(2)}\\ \vdash .(1).(2).*120·26.\supset \vdash .\text{Prop} \end{array} \]

The above proposition may also be deduced from *120·611, by *62·231.

*120·63. \(\vdash .\text{Cls induct} - \overleftarrow{{\in}}ʻ\Lambda \subset \text{Cls}^{2}\, \text{mult} \quad[*120·62.*88·2]\)

In virtue of this proposition the multiplicative axiom is not required in dealing with a finite number of factors, even when some or all of the factors are themselves infinite.

*120·64. \(\vdash : \text{Infin ax}.\lor.\text{Mult ax}\)

Dem. \[ \begin{array}{l} \vdash .*120·61.\text{Transp}.&\supset \vdash \colon\ldotp {\sim}\text{Infin ax}.\supset :\text{N}_{0}\text{C}\subset \text{NC induct}:\\ [*120·21] &\supset :(\kappa ).\kappa \in \text{Cls induct}:\\ [*120·62] &\supset :(\kappa ):\Lambda {\sim}\in \kappa .\supset .\exists !{\in}_{\Delta}ʻ\kappa :\\ [*88·37] &\supset :\text{Mult ax}\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

Thus of our two arithmetical axioms, the multiplicative axiom and the axiom of infinity, at least one must be true.

*120·7. \(\vdash :\alpha \in \text{Cls induct}.\alpha \subset \beta .\alpha \neq \beta .\supset .\text{Nc}ʻ\alpha < \text{Nc}ʻ\beta \quad[*120·426.*24·6]\)

*120·71. \(\vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \cup \sigma \in \text{Cls induct}.\equiv .\rho + \sigma \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·481.&\supset \vdash :\rho \cup \sigma \in \text{Cls induct}.\supset .\rho ,\sigma \in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .*120·481.&\supset \vdash :\rho ,\sigma \in \text{Cls induct}.\supset .\rho ,\sigma - \rho \in \text{Cls induct}.\\ [*120·21] & \supset .\text{N}_{0}\text{c}ʻ\rho ,\text{N}_{0}\text{c}ʻ(\sigma - \rho )\in \text{NC induct}.\\ [*120.45] &\supset .\text{N}_{0}\text{c}ʻ\rho +_{c} \text{N}_{0}\text{c}ʻ(\sigma - \rho )\in \text{NC induct}.\\ [*110·32] & \supset .\text{Nc}ʻ(\rho \cup \sigma )\in \text{NC induct}.\\ [*120·211] & \supset .\rho \cup \sigma \in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).(2). & \supset \vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \cup \sigma \in \text{Cls induct} &\qquad \text{(3)}\\ \vdash .*110·12.*120·214.&\supset \vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\\ &\downarrow (\Lambda \cap \sigma )ʻʻ\iota ʻʻ\rho ,(\Lambda \cap \rho )\downarrow ʻʻ\iota ʻʻ\sigma \in \text{Cls induct}.\\ [(3).(*110·01)] & \equiv .\rho + \sigma \in \text{Cls induct} &\qquad \text{(4)}\\ \vdash .(3).(4).\supset \vdash .\text{Prop} \end{array} \]

The above proposition is frequently used.

[Pg 230]

*120·72. \(\vdash :\rho ,\sigma \in \text{Cls induct}.\supset .\rho \times \sigma \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·21.\supset \vdash :\text{Hp}.&\supset .\text{N}_{0}\text{c}ʻ\rho ,\text{N}_{0}\text{c}ʻ\sigma \in \text{NC induct}.\\ [*120·5] &\supset .\text{Nc}ʻ(\rho \times \sigma )\in \text{NC induct}.\\ [*120·211] & \supset .\rho \times \sigma \in \text{Cls induct}:\supset \vdash .\text{Prop} \end{array} \]

*120·721. \(\vdash \colon\ldotp \exists !\rho .\exists !\sigma .\supset :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \times \sigma \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·512.*113·107.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :\rho \times \sigma \in \text{Cls induct}.\supset .\text{Nc}ʻ\rho ,\text{Nc}ʻ\sigma \in \text{NC induct}.\\ [*120·211] &\supset .\rho ,\sigma \in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .(1).*120·72.\supset \vdash .\text{Prop} \end{array} \]

*120·73. \(\vdash :\rho ,\sigma \in \text{Cls induct}.\supset .(\rho \,\,\text{exp}\,\,\sigma )\in \text{Cls induct} \quad[*120·52.*116·251]\)

*120·731. \[\begin{align}&\vdash \colon\ldotp \exists !\rho .\exists !\sigma .\rho {\sim}\in 1.\supset :\rho ,\sigma \in \text{Cls induct}.\equiv .(\rho \,\,\text{exp}\,\,\sigma )\in \text{Cls induct}\\ &[*120·56·561·73]\end{align}\]

*120·74. \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{Cl}ʻ\rho \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*116·72.*120·21.\supset \vdash :\text{Cl}ʻ\rho \in \text{Cls induct}.&\equiv .2^{\text{Nc}ʻ\rho }\cap tʻ\text{Cl}ʻ\rho \in \text{NC induct}.\\ [*120·123·52·56.*116·72.(*116·04)] &\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}.\\ [*120·21] &\equiv .\rho \in \text{Cls induct}.:\supset \vdash .\text{Prop} \end{array} \]

*120·741. \(\vdash :sʻ\kappa \in \text{Cls induct}.\supset .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*120·74. &\supset \vdash :\text{Hp}.\supset .\text{Cl}ʻsʻ\kappa \in \text{Cls induct}.\\ [*60·57.*120·481] &\supset .\kappa \in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .*40·13.*120·481.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\rho \in \kappa .\supset .\rho \in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*120·75. \(\vdash :sʻ\kappa \in \text{Cls induct}.\equiv .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*22·58. &\supset \vdash :\exists !\kappa -\text{Cls induct}.\supset .\exists !(\kappa \cup \iota ʻ\alpha )-\text{Cls induct} &\qquad \text{(1)}\\ \vdash .*120·71.*53·15.&\supset \vdash :sʻ\kappa \in \text{Cls induct}.\alpha \in \text{Cls induct}.\supset .\\ &sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct}:\\ [*5·6] &\supset \vdash \colon\ldotp sʻ\kappa \in \text{Cls induct}.\supset :\\ &\alpha {\sim}\in \text{Cls induct}.\lor.sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct}:\\ [*51·16]&\supset :\exists !(\kappa \cup \iota ʻ\alpha )-\text{Cls induct}.\lor.sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash \colon\ldotp \exists !\kappa -\text{Cls induct}.\lor.sʻ\kappa \in \text{Cls induct}:\supset :\\ &\exists !(\kappa \cup \iota ʻ\alpha )-\text{Cls induct}.\lor.sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(3)}\\ \vdash .*40·21.*120·212.&\supset \vdash .sʻ\Lambda \in \text{Cls induct} &\qquad \text{(4)}\\ \vdash .(3).(4).*120·26.&\supset \vdash \colon\ldotp \kappa \in \text{Cls induct}.\supset :\\ &\exists !\kappa -\text{Cls induct}.\lor.sʻ\kappa \in \text{Cls induct}\colon\ldotp \\ [*5·6]&\supset \vdash :\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}.\supset .sʻ\kappa \in \text{Cls induct} &\qquad \text{(5)}\\ \vdash .(5).*120·741.\supset \vdash .\text{Prop} \end{array} \]

[Pg 231]

*120·76. \(\vdash :\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*51·2. &\supset \vdash \colon\ldotp \alpha \in \kappa .\supset :\kappa =\kappa \cup \iota ʻ\alpha :\\ [*13·12] &\supset :{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .*83·41.*114·301. &\supset \vdash :\alpha {\sim}\in \kappa .\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\text{ sm }{\in}_{\Delta}ʻ\kappa \times \alpha &\qquad \text{(2)}\\ \vdash .(2).*120·214. &\supset \vdash \colon\ldotp \alpha {\sim}\in \kappa .\supset :\\ &{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct}.\equiv .{\in}_{\Delta}ʻ\kappa \times \alpha \in \text{Cls induct} &\qquad \text{(3)}\\ \vdash .(3).*120·72. &\supset \vdash \colon\ldotp \alpha {\sim}\in \kappa .\supset :\\ &{\in}_{\Delta}ʻ\kappa ,\alpha \in \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(4)}\\ \vdash .(1).(4). &\supset \vdash :{\in}_{\Delta}ʻ\kappa ,\alpha \in \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(5)}\\ \vdash .(5).*51·2.\text{Syll}. &\supset \vdash \colon\ldotp \kappa \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}:\supset :\\ &\kappa \cup \iota ʻ\alpha \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(6)}\\ \vdash .*83·15.*120·213. &\supset \vdash .{\in}_{\Delta}ʻ\Lambda \in \text{Cls induct}.\\ [\text{Simp}] &\supset \vdash :\Lambda \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\Lambda \in \text{Cls induct} &\qquad \text{(7)}\\ \vdash .(6).(7).*120·26. &\supset \vdash \colon\ldotp \kappa \in \text{Cls induct}.\supset :\\ &\kappa \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

The following propositions are concerned in establishing the converse of *120·76 subject to a suitable hypothesis. The final outcome is given in *120·77.

*120·761. \(\vdash :\exists !{\in}_{\Delta}ʻ\kappa .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\supset .\kappa \subset \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*83·41.*114·301. &\supset \vdash \colon\ldotp \alpha \in \kappa .\supset :{\in}_{\Delta}ʻ\kappa \text{ sm }\alpha \times {\in}_{\Delta}ʻ(\kappa -\iota ʻ\alpha ): &\qquad \text{(1)}\\ [*120·214] &\supset :{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\equiv .\alpha \times {\in}_{\Delta}ʻ(\kappa -\iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).*113·114. &\supset \vdash :\exists !{\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .\exists !\alpha .\exists !{\in}_{\Delta}ʻ(\kappa -\iota ʻ\alpha ) &\qquad \text{(3)}\\ \vdash .(2).(3).*120·721. & \supset \vdash \colon\ldotp \exists !{\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset :\\ &{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\supset .\alpha \in \text{Cls induct} &\qquad \text{(4)}\\ \vdash .(4).\text{Comm}.\supset \vdash .\text{Prop} \end{array} \]

*120·762. \(\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .{\sim}\exists !1\cap \kappa .\supset .(\exists R,S).R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} \)

Dem. \[ \begin{array}{l} \vdash .*51·2. & \supset \vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .\alpha \in \kappa .\supset .R,S\in {\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ).R\dot{\cap} S=\dot{\Lambda} &\qquad \text{(1)}\\ \vdash .*83·5.*55·201.\supset \\ &\vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .x,y\in \alpha .x\neq y.\alpha {\sim}\in \kappa .\supset .\\ &R\unicode{x228d} x\downarrow \alpha ,S\unicode{x228d} y\downarrow \alpha \in {\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ).(R\unicode{x228d} x\downarrow \alpha )\dot{\cap} (S\unicode{x228d} y\downarrow \alpha )=\dot{\Lambda} &\qquad \text{(2)}\\ \vdash .(1).(2).*52·41. &\supset \vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .\alpha \neq \Lambda .\alpha {\sim}\in 1.\supset .\\ &(\exists P,Q).P,Q\in {\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ).P\dot{\cap} Q=\dot{\Lambda} &\qquad \text{(3)}\\ \vdash .*51·16.&\supset \vdash \colon\ldotp \alpha =\Lambda .\lor.\alpha \in 1:\supset :\Lambda \in (\kappa \cup \iota ʻ\alpha ).\lor.\exists !1\cap (\kappa \cup \iota ʻ\alpha ) &\qquad \text{(4)}\\ \vdash .*22·58.&\supset \vdash \colon\ldotp \Lambda \in \kappa .\lor.\exists !1\cap \kappa :\supset :\Lambda \in (\kappa \cup \iota ʻ\alpha ).\lor.\exists !1\cap (\kappa \cup \iota ʻ\alpha ) &\qquad \text{(5)}\\ \vdash .(3).(4).(5).&\supset \vdash \colon\ldotp \Lambda \in \kappa .\lor.\exists !1\cap \kappa .\lor.(\exists R,S).R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} :\supset :\\ &\Lambda \in (\kappa \cup \iota ʻ\alpha ).\lor.\exists !1\cap (\kappa \cup \iota ʻ\alpha ).\lor.(\exists R,S).R,S\in {\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ).R\dot{\cap} S=\dot{\Lambda} &\qquad \text{(6)}\\ \vdash .*83·15.&\supset \vdash .(\exists R,S).R,S\in {\in}_{\Delta}ʻ\Lambda .R\dot{\cap} S=\dot{\Lambda} &\qquad \text{(7)}\\ \vdash .(6).(7).*120·26.\supset \vdash .\text{Prop} \end{array} \]

[Pg 232]

*120·764. \[\begin{align}&\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .{\sim}\exists !(1\cap \kappa ).\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \geq \text{Nc}ʻ\kappa \\ &[*120·762.*117·681]\end{align}\]

*120·765. \[\begin{align}\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .{\sim}&\exists !(1\cap \kappa ).\kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ\lambda .\supset .\\ &\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ\kappa \quad[*120·762.*117·684]\end{align}\]

*120·766. \[\begin{align}\vdash :\lambda {\sim}\in \text{Cls induct}.\Lambda {\sim}\in \lambda.{\sim}\exists !(1\cap \lambda ).&\exists !{\in}_{\Delta}ʻ\lambda .\supset .\\ &\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda {\sim}\in \text{NC induct}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*120·491.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset .\\ &(\exists \kappa ).\kappa \subset \lambda .\text{Nc}ʻ\kappa =\nu .\Lambda {\sim}\in \kappa .{\sim}\exists !(1\cap \kappa ).\\ [*120·765] &\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \nu :\\ [*120·121] &\supset :\nu \in \text{NC induct}.\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \nu +_{c}1.\\ [*120·429] &\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda >\nu :\\ [*117·42] &\supset :\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda {\sim}\in \text{NC induct}\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*120·767. \[\begin{align}&\vdash :{\in}_{\Delta}ʻ\lambda \in \text{Cls induct}.\Lambda {\sim}\in \lambda .{\sim}\exists !(1\cap \lambda ).\exists !{\in}_{\Delta}ʻ\lambda .\supset .\lambda \in \text{Cls induct}\\ &[*120·766.\text{Transp}]\end{align}\]

*120·77. \[\begin{align}\vdash \colon\ldotp \Lambda {\sim}\in \kappa .{\sim}&\exists !(1\cap \kappa ).\exists !{\in}_{\Delta}ʻ\kappa .\supset :\\ &{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\equiv .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\\ &[*120·76·761·767]\end{align}\]

Summary of *121.

The present number is concerned with the class of terms between \(x\) and \(y\) with respect to some relation \(P\), i.e. those terms which lie on a road from \(x\) to \(y\) on which any two consecutive terms have the relation \(P\). Such a road may be called a \(P\)-road, and if \(zPw\), the step from \(z\) to \(w\) may be called a \(P\)-step. In order that a \(P\)-road from \(x\) to \(y\) should exist, it is necessary and sufficient that we should have \(xP_{\text{po}}y\). When this condition is fulfilled, there will in general be many \(P\)-roads from \(x\) to \(y\). But if \(P\in \text{Cls}\rightarrow 1.{\sim}(yP_{\text{po}}y)\), or if \(P\in 1\rightarrow \text{Cls}.{\sim}(xP_{\text{po}}x)\),then at most one road leads from \(x\) to \(y\). This follows from the propositions of *96. In virtue of those propositions, if \(P\in \text{Cls}\rightarrow 1.{\sim}(yP_{\text{po}}y).xP_{\text{po}}y\), \(P\) is \(1\rightarrow 1\) throughout the road from \(x\) to \(y\), and this road forms an open series. The two other possibilities with a \(\text{Cls}\rightarrow 1\) are (assuming \(xP_{\text{po}}y\)) \[ \begin{aligned} &(1)\quad xP_{\text{po}}x,\\ &(2) \quad yP_{\text{po}}y.{\sim}(xP_{\text{po}}x). \end{aligned} \]

In the first case, there is a cyclic road from \(x\) to \(x\), and there are two roads from \(x\) to \(y\), one consisting of that part of the cycle which is required to reach \(y\), the other consisting of this part together with the whole cycle required to travel from \(y\) back to \(y\). Thus the class of terms which can be reached in some journey from \(x\) to \(y\) is the whole class of descendants of \(x\), i.e. the class \(\overleftarrow{R}_{*}ʻx\), which is the cycle composing the road from \(x\) to \(x\).

In the second case, the descendants of \(x\) form a \(Q\), and \(y\) is in the circular part of the \(Q\). Here, as before, there are two roads from \(x\) to \(y\), of which the first stops as soon as it reaches \(y\), while the second proceeds to travel round the circle until it comes to \(y\) again. Thus here again, all the descendants of \(x\) lie on some road between \(x\) and \(y\).

The interval between \(x\) and \(y\) is defined as the class of terms lying on some road from \(x\) to \(y\). There will be four kinds of interval, according as we do or do not include the end-points as such. We denote the kind including both end-points by \[ P(x\vdash\dashv y), \] that excluding both by \[ P(x\unicode{x2013} y) \] and the other two respectively by \[ P(x\dashv y), P(x \unicode{x27dd} y). \]

[Pg 234]

The definitions are \[ \overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{*}ʻy, \quad\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy, \quad\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{*}ʻy, \quad\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy. \]

If \(P\) is either one-many or many-one, it will be one-one throughout the interval \(P(x\vdash\dashv y)\), except at most at one exceptional point, namely the junction of the tail and circle of the \(Q\). If \(xP_{\text{po}}x\) or \({\sim}(yP_{\text{po}}y)\), the interval between \(x\) and \(y\) cannot be \(Q\)-shaped, but must be either open or cyclic; in either case, \(P\) is \(1\rightarrow 1\) throughout \(P(x\vdash\dashv y)\), with no exceptions; for if \(P\in \text{Cls}\rightarrow 1\), \(P\) is \(1\rightarrow 1\) throughout the interval because the interval is contained in \(\overleftarrow{P}_{*}ʻx\), and if \(P\in 1\rightarrow \text{Cls}\), because the interval is contained in \(\overrightarrow{P}_{*}ʻy\). Thus throughout this number we shall constantly have the hypothesis \(P\in(\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls})\); if \(P\in\text{Cls}\rightarrow 1\), the interval is to be supposed traversed from \(x\) to \(y\), while if \(P\in 1\rightarrow \text{Cls}\), it is to be supposed traversed from \(y\) to \(x\). In either case the interval between \(x\) and \(y\) must be an inductive class. This is proved in *121·47. If, however, \(P\) is serial (cf. *204), and thus neither many-one nor one-many, the interval between \(x\) and \(y\) is the stretch of the series between \(x\) and \(y\), with or without end-points according to the definition chosen, and need not be an inductive class.

If the interval between \(x\) and \(y\) (both included) has \(\nu +_{c}1\) members, we say that \(xP_{\nu }y\). Thus if there is only one road from \(x\) to \(y\), "\(xP_{\nu }y\)" means that it requires \(\nu\) steps to get from \(x\) to \(y\). Assuming \(P\in \text{Cls}\rightarrow 1\), if we also have \(P_{\text{po}}\,\unicode{x2abd}\, J\) (i.e. if none of the families of \(P\) are cyclic), then if \(xP_{\nu }y\) and \(yPz\), we shall have \(xP_{\nu +_{c}1}z\). On this basis an inductive theory of \(P_{\nu}\) is built up, and it is shown that the class of such relations as \(P_{\nu }\) for different inductive values of \(\nu\) is the same as \(\text{Potid}ʻP\), the class of powers of \(P\) including \(I\upharpoonright CʻP\) (*121·5). The definition of \(P_{\nu }\) is \[ P_{\nu } = \hat{x} \hat{y} \{\text{N}_{0}\text{c}ʻP(x\vdash\dashv y)=v+_{c}1\} \quad\text{Df}. \]

The whole class of such relations as \(P_{\nu}\) for different inductive values of \(\nu\) is called \(\text{finid}ʻP\), i.e. we put \[ \text{finid}ʻP=\hat{R} \{(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ\Lambda .R=P_{\nu }\} \quad\text{Df}. \]

If \(BʻP\) exists, and if \(P\in \text{Cls}\rightarrow 1\), then the descendants of \(BʻP\), so long as we do not reach a term \(y\) for which \(yP_{\text{po}}y\), may be unambiguously described as the 2nd, 3rd, ... \(\nu\)th, ... terms of the posterity of \(BʻP\), \(BʻP\) itself being the 1st term. The correlation thus effected with the inductive cardinals is the logical essence of the process of counting; the last cardinal used in the correlation is the cardinal number of terms counted. We will call these terms \(1_{P}\), \(2_{P}\), ... \(\nu_{P}\), ..., defining \(\nu _{P}\) as follows: \[ \nu _{P} = \breve{P} _{\nu -_{c}1}ʻBʻP \quad\text{Df}, \]

This notation does not conflict with \(\nu _{\xi }\) as defined in *65.01. There \(\xi\) must be a class if \(\nu\) is a cardinal, here \(\nu\) must be a cardinal and \(P\) a relation.

[Pg 235]

Hence whenever \(\nu _{P}\) exists, the number of terms from the beginning to \(\nu _{P}\) (both included) is \(\nu\). This is the fact upon which counting relies. If \(P\) is a many-one and \(P_{\text{po}}\) is contained in diversity, and \(\nu\) is any inductive cardinal other than 0, then \(\nu _{P}\) exists when and only when \(\overleftarrow{P}_{*}ʻBʻP\) has at least \(\nu\) members; i.e. roughly speaking, \(\nu _{P}\) exists whenever it could possibly be expected to exist. In this case the whole posterity of \(BʻP\) is contained in the series \(1_{P}\), \(2_{P}\), ... \(\nu_{P}\), ... (*121·62). If the posterity is an inductive class, this series stops; if not, it forms a progression (cf. *122).

The propositions of the present number are very useful, not only in this section, but in the ordinal theory of finite and infinite and in parts of the book subsequent to that theory.

After some propositions which merely repeat definitions and give immediate consequences, we proceed (*121·3 ff.) to the theory of \(P_{\nu }\). We have

*121·302. \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{0}=I\upharpoonright CʻP\)

*121·305. \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}\,\unicode{x2abd}\, P\)

*121·31. \(\vdash :P\in (l\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}=P\)

When \(P\) is a transitive serial relation, we shall have \(P_{1}=P\dot{-} P^{2}\).

*121·321. \(\vdash :\nu >0.\supset .P_{\nu }\,\unicode{x2abd}\, P_{\text{po}}\)

*121·333. \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{\nu +_{c}1}=P\mid P_{\nu }\)

*121·35·351·352. \[\begin{align}\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).&P_{\text{po}}\,\unicode{x2abd}\, J.\mu ,\nu \in \text{NC induct}.\supset .\\ &P_{\mu }\mid P_{\nu }=P_{\nu }\mid P_{\mu }=P_{\mu +_{c}\nu }\end{align}\]

A similar result holds for \((P_{\mu })_{\nu }\), which = \(P_{\mu x_{c}\nu }\) n the same circumstances.

We next proceed to the proof that an interval (under a similar hypothesis) is always an inductive class. This occupies *121·4—·47, being summed up in the proposition

*121·47. \(\vdash :R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).\supset .R(x \vdash\dashv z)\in \text{Cls induct}\)

This is an important proposition. It leads to

*121·481. \[\begin{align}\vdash \colon\ldotp R\in \text{Cls}\rightarrow 1.\supset :\text{Nc}ʻR(x\vdash\dashv y)\leq \text{Nc}ʻ&R(x \vdash\dashv z).\equiv .\\ &R(x \vdash\dashv y)\subset R(x \vdash\dashv z)\end{align}\]

with a similar proposition if \(R\in 1\rightarrow \text{Cls}\).

The next set of propositions (*121·5—·52) is concerned with \(\text{finid}ʻP\). Assuming \(P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, J\), we prove that \(\text{finid}ʻP=\text{Potid}ʻP\) and \(\text{finid}ʻP-\iotaʻP_{0}\subset \text{Pot}ʻP\) (*121·5); that if \(P\) is not null, \(\text{finid}ʻP-\iota ʻP_{0}=\text{Pot}ʻP\) (*121·501); that \(\dot{s}ʻ\text{finid}ʻP=P_{*}\) (*121·52) and \(\dot{s} ʻ(\text{finid}ʻP-\iotaʻP_{0})=P_{\text{po}}\) (*121·502); and that \(P_{2}=P^{2}.P_{3}=P^{3}\) etc. (*121·51).

[Pg 236]

Our next set of propositions is concerned with \(\nu _{P}\) (*121·6—·638). We have

*121·601. \(\vdash : \text{E}! BʻP . \supset . BʻP = 1_{P} . {\sim} \{(BʻP) P_{\text{po}} (BʻP)\}\)

*121·602. \(\vdash : \text{E}! BʻP . P \in 1 \rightarrow 1 . \supset . \breve{P} ʻBʻP = 2_{P}\)

*121·634. \[\begin{align}\vdash \colon\ldotp P \in \text{Cls} \rightarrow 1 . P_{\text{po}} \,\unicode{x2abd}\, J . \nu \in &\text{NC induct} - \iota ʻ0 . \supset :\\ &\nu _{P} \in \text{D} ʻP . \equiv . \text{E}! (\nu +_{c} 1)_{P}\end{align}\]

Finally we have three propositions (*121·7—*121·72) on \(\overrightarrow{R}_{*}ʻx\), of which the most useful is

*121·7. \(\vdash : R \in 1 \rightarrow 1 . aBR . aR_{*}x . \supset . \overrightarrow{R}_{*}ʻx = R(a \vdash\dashv x) . \overrightarrow{R}_{*}ʻx \in \text{Cls induct}\)

*121·01. \(P(x \unicode{x2013} y) = \overleftarrow{P}_{\text{po}}ʻx \cap \overrightarrow{P}_{\text{po}}ʻy \quad\text{Df}\)

*121·011. \(P(x \dashv y) = \overleftarrow{P}_{\text{po}} ʻx \cap \overrightarrow{P}_{*} ʻy \quad\text{Df}\)

*121·012. \(P(x \unicode{x27dd} y) = \overleftarrow{P}_{*} ʻx \cap \overrightarrow{P}_{\text{po}} ʻy \quad\text{Df}\)

*121·013. \(P(x \vdash\dashv y) = \overleftarrow{P}_{*} ʻx \cap \overrightarrow{P}_{*} ʻy \quad\text{Df}\)

*121·02. \(P_\nu = \hat{x} \hat{y} \{\text{N}_0\text{c} ʻP (x \vdash\dashv y) = \nu +_{c} 1\} \quad\text{Df}\)

*121·03. \(\text{finid} ʻP = \hat{R} \{(\exists \nu ) . \nu \in \text{NC induct} - \iota ʻ\Lambda . R = P_{\nu }\} \quad\text{Df}\)

*121·031. \(\text{fin} ʻP = \hat{R} \{(\exists \nu ). \nu \in \text{NC induct} - \iota ʻ\Lambda - \iota ʻ0 . R = P_{\nu }\} \quad\text{Df}\)

*121·04. \(\nu _{P} = \breve{P} _{\nu -_{c} 1} ʻBʻP \quad\text{Df}\)

*121·1. \(\vdash : z \in P (x \unicode{x2013} y) . \equiv . xP_{\text{po}}z . zP_{\text{po}}y \quad[(*121·01)]\)

*121·101. \(\vdash : z \in P (x \dashv y) . \equiv . xP_{\text{po}}z . zP_{*}y\)

*121·102. \(\vdash : z \in P (x \unicode{x27dd} y) . \equiv . xP_{*}z . zP_{\text{po}}y\)

*121·103. \(\vdash : z \in P (x \vdash\dashv y) . \equiv . xP_{*}z . zP_{*}y\)

*121·11. \(\vdash : xP_{\nu }y . \equiv . \text{N}_0\text{c} ʻP (x \vdash\dashv y) = \nu +_{c} 1 \quad[(*121·02)]\)

*121·12. \(\vdash : R \in \text{finid} ʻP . \equiv . (\exists \nu ) . \nu \in \text{NC induct} - \iota ʻ\Lambda . R = P_\nu \quad[(*121·03)]\)

*121·121. \(\vdash : R \in \text{fin} ʻP . \equiv . (\exists \nu ) . \nu \in \text{NC induct} - \iota ʻ\Lambda - \iota ʻ0 . R = P_\nu \quad[(*121·031)]\)

*121·13. \(\vdash : f (\nu _{P}) . \equiv . f(\breve{P} _{\nu -_{c} 1} ʻBʻP) \quad[(*121·04)]\)

*121*131. \(\vdash : \text{E}! \breve{P} _{\nu -_{c} 1} ʻBʻP . \supset . \nu _{P} = \breve{P} _{\nu -_{c} 1}ʻBʻP \quad[*121·13.*14·28]\)

*121·14. \(\vdash . P (x \unicode{x2013} y) = \breve{P} (y \unicode{x2013} x) \quad[*121·1 . *91·53]\)

*121·141. \(\vdash . P (x \dashv y) = \breve{P} (y \unicode{x27dd} x)\)

*121·142. \(\vdash . P (x \unicode{x27dd} y) = \breve{P} (y \dashv x)\)

*121·143. \(\vdash . P (x \vdash\dashv y) = \breve{P} (y \vdash\dashv x)\)

*121·2. \(\vdash : {\sim} (xP_{\text{po}}x) . \supset . x {\sim} \in P (x \unicode{x2013} y) \quad[*121·1]\)

[Pg 237]

*121·201. \(\vdash : {\sim} (yP_{\text{po}}y) . \supset . y {\sim} \in P (x \unicode{x2013} y)\)

*121·202. \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .x,y{\sim}\in P(x\unicode{x2013}y) \quad[*121·2·201]\)

*121*21. \(\vdash :xP_{\text{po}}y.\equiv .y\in P(x\dashv y).\equiv .\exists !P(x\dashv y)\)

Dem. \[ \begin{array}{l} \vdash .*90·12.*91·54.\supset \vdash :xP_{\text{po}}y.&\equiv .xP_{\text{po}}y.yP_{*}y.\\ [*121·101] &\equiv .y\in P(x\dashv y) &\qquad \text{(1)}\\ \vdash .*121·101.\supset \vdash :\exists !P(x\dashv y).&\equiv .xP_{\text{po}}\mid P_{*}y.\\ [*91·574] &\equiv .xP_{\text{po}}y &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·22. \(\vdash :xP_{\text{po}}y.\equiv .x\in P(x\unicode{x27dd} y).\equiv .\exists !P(x\unicode{x27dd} y)\)

*121·23. \(\vdash :xP_{*}y.\equiv .x,y\in P(x\vdash\dashv y).\equiv .\exists !P(x\vdash\dashv y)\)

*121·231. \(\vdash :x\in CʻP.\equiv .x\in P(x\vdash\dashv x).\equiv .\exists !P(x\vdash\dashv x) \quad[*121·23.*90·12]\)

*121·24. \(\vdash :xP_{\text{po}}y.\supset .P(x\dashv y)=P(x\unicode{x2013}y)\cup \iota ʻy\)

Dem. \[ \begin{array}{l} \vdash .*91·54.*121·101.\supset \\ \vdash \colon\ldotp z\in P(x\dashv y). &\equiv :xP_{\text{po}}z:zP_{\text{po}}y.\lor.z=y.y\in CʻP:\\ [*13·193.*91·504]&\equiv:xP_{\text{po}}z.zP_{\text{po}}y.\lor.xP_{\text{po}}y.z=y &\qquad \text{(1)}\\ \vdash .(1).*4·73.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp z\in P(x\dashv y).&\equiv :xP_{\text{po}}z.zP_{\text{po}}y.\lor.z=y\colon\colon \supset \vdash .\text{Prop} \end{array} \]

*121·241. \(\vdash :xP_{\text{po}}y.\supset .P(x \unicode{x27dd} y)=P(x\unicode{x2013}y)\cup \iota ʻx\)

*121·242. \[\begin{align}\vdash :xP_{*}y.\supset .P(x\vdash\dashv y)&=P(x\dashv y)\cup \iota ʻx=P(x \unicode{x27dd} y)\cup \iota ʻy\\ &=P(x-y)\cup \iota ʻx\cup \iota ʻy\end{align}\]

*121*25. \(\vdash .P_{\text{po}}(x\unicode{x2013}y)=P(x\unicode{x2013}y) \quad[*91·601.*121·1]\)

*121·251. \(\vdash .P_{\text{po}}(x\dashv y)=P(x\dashv y)\)

*121·252. \(\vdash .P_{\text{po}}(x\unicode{x27dd} y)=P(x \unicode{x27dd} y)\)

*121·253. \(\vdash .P_{\text{po}}(x\vdash\dashv y)=P(x\vdash\dashv y)\)

*121·254. \(\vdash .P_{\nu }=(P_{\text{po}})_{\nu } \quad[*121·253·11]\)

*121·254 is frequently used in the theory of series.

*121·26. \(\vdash .\breve{P} _{\nu }=(\breve{P} )_{\nu }\)

Dem. \[ \begin{array}{l} \vdash .*121·11·143.\supset \vdash :x\breve{P} _{\nu }y.&\equiv .\text{N}_{0}\text{c}ʻ\breve{P} (x\vdash\dashv y)=\nu +_{c}1.\\ [*90·132.*121·11] &\equiv .x(\breve{P} )_{\nu }y:\supset \vdash .\text{Prop} \end{array} \]

*121·27. \(\vdash :xP_{\nu }y.\supset .\nu ,\nu +_{c}1\in \text{NC}-\iota ʻ\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*121·11.*103·12.\supset \vdash :\text{Hp}.\supset .P(x\vdash\dashv y)\in \nu +_{c}1 &\qquad \text{(1)}\\ \vdash .(1).*110·4·42.\supset \vdash .\text{Prop} \end{array} \]

[Pg 238]

*121·271. \(\vdash :{\sim}(\nu ,\nu +_{c}1\in \text{NC}-\iota ʻ\Lambda ).\supset .P_{\nu }=\dot{\Lambda} \quad[*121·27.\text{Transp}]\)

*121·272. \(\vdash :\dot{\exists} !P_{\nu }.\supset .\nu \geq 0.\nu +_{c}1>0.\nu +_{c}1\geq 1\)

Dem. \[ \begin{array}{l} \vdash .*117·5.*121·27.\supset \vdash :\text{Hp}.&\supset .\nu \geq 0. &\qquad \text{(1)}\\ [*117·561.*110·641] &\supset .\nu +_{c}1\geq 1. &\qquad \text{(2)}\\ [*117·511·531] &\supset .\nu +_{c}1>0 &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*121·273. \(\vdash :\dot{\exists} !P_{\nu +_{c}1}.\supset .\nu +_{c}1>0\)

Dem. \[ \begin{array}{l} \vdash .*121·27.*110·4.\supset \vdash :\text{Hp}.&\supset .\nu \in \text{NC}-{℩}ʻ\Lambda .\\ [*117·6] &\supset .\nu +_{c}1\geq 1.\\ [*117·511·531] &\supset .\nu +_{c}1>0:\supset \vdash .\text{Prop} \end{array} \]

*121·3. \(\vdash .P_{0}\,\unicode{x2abd}\, 1\upharpoonright CʻP\)

Dem. \[ \begin{array}{l} \vdash .*121·11.\supset \vdash :xP_{0}y.&\equiv .P(x\vdash\dashv y)\in 1.\\ [*121·23] &\supset .xP_{*}y.x=y.\\ [*90·12] &\supset .x(I\upharpoonright CʻP)y:\supset \vdash .\text{Prop} \end{array} \]

*121·301. \(\vdash :{\sim}(xP_{\text{po}}x).\supset :xP_{0}y.\equiv .x\in CʻP.x=y\)

Dem. \[ \begin{array}{l} \vdash .*91·542·56. &\supset \vdash :xP_{*}z.zP_{*}x.x\neq z.\supset .xP_{\text{po}}x &\qquad \text{(1)}\\ \vdash .(1).\text{Transp}. \supset \vdash \colon\ldotp \text{Hp}.&\supset :xP_{*}z.zP_{*}x.\supset _{x,z}.x=z:\\ [*121·231] &\supset :x\in CʻP.\supset .P(x\vdash\dashv x)={℩}ʻx:\\ [*13·12.*52·22] &\supset :x\in CʻP.x=y.\supset .P(x\vdash\dashv y)\in 1.\\ [*121·11] &\supset .xP_{0}y &\qquad \text{(2)}\\ \vdash .(2).*121·3.\supset \vdash .\text{Prop} \end{array} \]

*121·302. \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{0}=I\upharpoonright CʻP \quad[*121·301]\)

*121·303. \(\vdash :\text{Nc}ʻP(x\vdash\dashv y)>1.\supset .xP_{\text{po}}y\)

Dem. \[ \begin{array}{l} \vdash .*121·23.*52·22.*117·42.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in P(x\vdash\dashv y).P(x\vdash\dashv y)\neq {℩}ʻx:\\ [*51·4.\text{Transp}] &\supset :(\exists z).z\neq x.z\in P(x\vdash\dashv y):\\ [*121·103.*91·542] &\supset :(\exists z).xP_{\text{po}}z.zP_{*}y:\\ [*91·574] &\supset :xP_{\text{po}}y\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*121·304. \(\vdash \colon\ldotp P_{\text{po}}\,\unicode{x2abd}\, J.\supset :xP_{1}y.\equiv .P(x\vdash\dashv y)={℩}ʻx\cup {℩}ʻy.x\neq y\)

Dem. \[ \begin{array}{l} \vdash .*121·303·11.\supset \vdash :\text{Hp}.xP_{1}y.&\supset .xP_{\text{po}}y.\\ [\text{Hp}] &\supset .x\neq y &\qquad \text{(1)}\\ \vdash .(1).*54·53·101.*121·23·11.&\supset \vdash .\text{Prop} \end{array} \]

[Pg 239]

*121·305. \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}\,\unicode{x2abd}\, P\)

Dem. \[ \begin{array}{l} \vdash .*121·303.\supset \vdash :\text{Hp}.xP_{1}y.&\supset .xP_{\text{po}}y.\\ [*91·52] &\supset .(\exists z).xPz.zP_{*}y &\qquad \text{(1)}\\ \vdash .*121·304.*91·542.\supset \\ \vdash \colon\ldotp \text{Hp}.xP_{1}y.&\supset :xP_{\text{po}}z.zP_{*}y.\supset .z=y:\\ [*91*502] &\supset :xPz.zP_{*}y.\supset .z=y &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·306. \(\vdash :P\in 1\rightarrow \text{Cls}.{\sim}(xP_{\text{po}}x).xPy.\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy.x\neq y\)

Dem. \[ \begin{array}{l} \vdash .*91·542. &\supset \vdash :xP_{*}z.zP_{*}y.z\neq x.z\neq y.xPy.\supset :xP_{\text{po}}z.zP_{\text{po}}y.xPy:\\ [*34·1] &\supset :xP_{\text{po}}z.zP_{\text{po}}\mid \breve{P} x:\\ [*92·11] &\supset :P\in 1\rightarrow \text{Cls}.\supset .xP_{\text{po}}z.zP_{*}x:\\ [*91·574] &\supset :P\in 1\rightarrow \text{Cls}.\supset .xP_{\text{po}}x &\qquad \text{(1)}\\ \vdash .(1).\text{Transp}.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xP_{*}z.zP_{*}y.\supset _{z}:z=x.\lor.z=y &\qquad \text{(2)}\\ \vdash .*121·23. &\supset \vdash :\text{Hp}.\supset .x,y\in P(x\vdash\dashv y) &\qquad \text{(3)}\\ \vdash .*91·502.& \supset \vdash :\text{Hp}.\supset .x\neq y &\qquad \text{(4)}\\ \vdash .(2).(3).(4).*121·103.\supset \vdash .\text{Prop} \end{array} \]

*121·307. \[\begin{align}&\vdash :P\in \text{Cls}\rightarrow 1.{\sim}(yP_{\text{po}}y).xPy.\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy.x\neq y\\ &[*121·306·143]\end{align}\]

*121·308. \[\begin{align}&\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P\,\unicode{x2abd}\, P_{1}\\ &[*121·306·307·11.*54·101]\end{align}\]

*121·31. \(\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}=P \quad[*121·305·308]\)

*121·32. \(\vdash .P_{\nu }\,\unicode{x2abd}\, P_{*}\)

Dem. \[ \begin{array}{l} \vdash .*121·11.*120·421.*101·14.\text{Transp}.&\supset \vdash :xP_{\nu }y.\supset .\exists !P(x\vdash\dashv y).\\ [*121·23] &\supset .xP_{*}y:\supset \vdash .\text{Prop} \end{array} \]

If \(\nu\) is not a cardinal, or if \(\nu +_{c}1=\Lambda\), \(P_{\nu}=\dot{\Lambda}\).

*121·321. \(\vdash :\nu >0.\supset .P_{\nu }\,\unicode{x2abd}\, P_{\text{po}}\)

Dem. \[ \begin{array}{l} \vdash .*120·428.*121·11.\supset \vdash :\text{Hp}.xP_{\nu }y.&\supset .\text{Nc}ʻP(x\vdash\dashv y)>1.\\ [*117·55.*52·181.*121·23] &\supset .(\exists z).z\in P(x\vdash\dashv y).z\neq x.\\ [*121·103.*91·542] &\supset .(\exists z).xP_{\text{po}}z.zR_{*}y.\\ [*91·574] &\supset .xP_{\text{po}}y:\supset \vdash .\text{Prop} \end{array} \]

*121·322. \(\vdash .CʻP_{\nu }\subset CʻP \quad[*121·32.*90·14]\)

[Pg 240]

*121·323. \(\vdash :\nu >0.\supset .\text{D}ʻP_{\nu }\subset \text{D}ʻP.\text{ᗡ}ʻP_{\nu }\subset \text{ᗡ}ʻP \quad[*121·321.*91·504]\)

*121·324. \(\vdash .\text{D}ʻP_{\nu +_{c}1}\subset \text{D}ʻP.\text{ᗡ}ʻP_{\nu +_{c}1}\subset \text{ᗡ}ʻP\)

Dem. \[ \begin{array}{l} \vdash .*121·273·323.\supset \vdash :\dot{\exists} !P_{\nu +_{c}1}.\supset .\text{D}ʻP_{\nu +_{c}1}\subset \text{D}ʻP.\text{ᗡ}ʻP_{\nu +_{c}1}\subset \text{ᗡ}ʻP &\qquad \text{(1)}\\ \vdash .(1).*33·241.\supset \vdash .\text{Prop} \end{array} \]

*121·325. \(\vdash :\dot{\exists} !P_{\mu }\dot{\cap} P_{\nu }.\supset .\mu =\nu\)

Dem. \[ \begin{array}{l} \vdash .*121·11.&\supset \vdash :\text{Hp}.\supset .\exists !(\mu +_{c}1)\cap (\nu +_{c}1)\cap t_{0}ʻ\mu .\\ [*100·42.*110·4] &\supset .\exists !(\mu +_{c}1)\cap t_{0}ʻ\mu .(\mu +_{c}1)\cap t_{0}ʻ\mu =\nu +_{c}1.\\ [*120·311] &\supset .\mu =\nu :\supset \vdash .\text{Prop} \end{array} \]

*121·326. \(\vdash .\text{fin}ʻP\subset \text{finid}ʻP.\text{finid}ʻP-\iota ʻP_{0}\subset \text{fin}ʻP \quad[*121·12·121]\)

*121·327. \(\vdash :\dot{\exists} !P_{0}.\supset .\text{fin}ʻP=\text{finid}ʻP-\iota ʻP_{0}\)

Dem. \[ \begin{array}{l} \vdash .*121·325.\text{Transp}.*121·121.\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in \text{fin}ʻP.\supset .R\neq P_{0} &\qquad \text{(1)}\\ \vdash .(1).*121·326.\supset \vdash .\text{Prop} \end{array} \]

*121·33·331 are lemmas for *121·332, which is a very useful proposition.

*121·33. \[\begin{align}\vdash \colon\ldotp P\in 1\rightarrow \text{Cls}.\supset :&z\in P(x-y).\equiv .z\in P(x\dashv Pʻy):\\ &z\in P(x \unicode{x27dd} y).\equiv .z\in P(x\vdash\dashv Pʻy)\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*71·7.\supset \vdash \colon\ldotp \text{Hp}.\supset :zP_{*}(Pʻy).&\equiv .zP_*\mid Py.\\ [*91·52] &\equiv .zP_{\text{po}}y &\qquad \text{(1)}\\ \vdash .(1).*121·1·101·102·103.\supset \vdash .\text{Prop} \end{array} \]

From the above proposition it follows that \[ P\in 1\rightarrow \text{Cls}.y\in \text{ᗡ}ʻP.\supset .P(x-y)=P(x\dashv Pʻy).P(x \unicode{x27dd} y)=P(x\vdash\dashv Pʻy). \]

This does not follow unless \(y\in \text{ᗡ}ʻP\), because \[ \begin{aligned} P&(x\unicode{x2013}y)=P(x\dashv Pʻy).\supset .\text{E}!Pʻy,\\ \text{whereas}\qquad &z\in P(x\unicode{x2013}y).\equiv _{z}.z\in P(x\dashv Pʻy) \end{aligned} \] will always be true if \(y{\sim}\in \text{ᗡ}ʻP\), and therefore (when \(P\in 1\rightarrow \text{Cls}\)) if \({\sim}\text{E}!Pʻy\).

*121·331. \(\vdash \colon\ldotp P \in 1\rightarrow \text{Cls}.P_{\text{po}}\,\unicode{x2abd}\, J.\supset :xP_{\nu }(Pʻy).\equiv .xP_{\nu +_{c}1}y\)

Dem. \[ \begin{array}{l} \vdash .*121·324.*71·16. &\supset \vdash \colon\ldotp \text{Hp}.\supset :xP_{\nu +_{c}1}y.\supset .\text{E}!Pʻy &\qquad \text{(1)}\\ \vdash .*121·33. &\supset \vdash :\text{Hp}.\text{E}!Pʻy.\supset .P(x \unicode{x27dd} y)=P(x\vdash\dashv Pʻy) &\qquad \text{(2)}\\ \vdash .*121·242·32.(2). &\supset \vdash :\text{Hp}(2).xP_{*}y.\supset .P(x\vdash\dashv y)=P(x\vdash\dashv Pʻy)\cup \iota ʻy &\qquad \text{(3)}\\ \vdash .*91·52. \supset \vdash :\text{Hp}.&\supset .{\sim}(yP_{*}\mid Py).\\ [*71·7] &\supset .{\sim}\{yP_{*} (Pʻy)\} .\\ [*121·103] &\supset .{\sim}\{y\in P(x\vdash\dashv Pʻy)\} &\qquad \text{(4)}\\ \vdash .(3).(4).*110·63.&\supset \vdash :\text{Hp}(3).\supset .\text{Nc}ʻP(x\vdash\dashv y)=\text{Nc}ʻP(x\vdash\dashv Pʻy)+_{c}1 &\qquad \text{(5)}\\ \vdash .(1).(5).*121·11·32.\supset \\ \vdash :\text{Hp}.xP_{\nu +_{c} 1}y.&\supset .(\nu +_{c} 1) +_{c} 1 = \text{Nc}ʻP(x\vdash\dashv Pʻy)+_{c} 1.\\ [*120·311.*121·27] &\supset . \nu +_{c} 1 = \text{Nc}ʻP(x \vdash\dashv Pʻy).\\ [*121·11] &\supset .xP_{\nu } (Pʻy) &\qquad \text{(6)}\\ \vdash .(5).*14·21.*121·11·32. &\supset \vdash : \text{Hp}.xP_{\nu } (Pʻy).\supset .\text{Nc}ʻP(x \vdash\dashv y)=(\nu +_{c} 1) +_{c} 1.\\ [*121·11] &\supset . xP_{\nu +_{c} 1}y &\qquad \text{(7)}\\ \vdash .(6).(7).\supset \vdash . \text{Prop} \end{array} \]

[Pg 241]

*121·332. \(\vdash : P \in 1 \rightarrow \text{Cls}. P_{\text{po}} \,\unicode{x2abd}\, J. \supset . P_{\nu +_{c} 1} = P_{\nu } \mid P \quad[*121·331]\)

*121·333. \(\vdash : P \in \text{Cls} \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \supset . P_{\nu +_{c} 1} = P \mid P_{\nu }\)

*121·34. \(\vdash : P \in 1 \rightarrow \text{Cls}. P_{\text{po}} \,\unicode{x2abd}\, J. \nu \in \text{NC induct}. \supset . P_{\nu } \in 1 \rightarrow \text{Cls}\)

Dem. \[ \begin{array}{l} \vdash . *121·3. &\supset \vdash . P_{0} \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ \vdash . *121·332. &\supset \vdash \colon\ldotp \text{Hp}. \supset : P_{\nu } \in 1 \rightarrow \text{Cls}. \supset . P_{\nu +_{c} 1} \in 1 \rightarrow \text{Cls} &\qquad \text{(2)}\\ \vdash . (1).(2). *120·11.\supset \vdash . \text{Prop} \end{array} \]

*121·341. \(\vdash : P \in \text{Cls} \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \nu \in \text{NC induct}. \supset . P_{\nu } \in \text{Cls} \rightarrow 1\)

*121·342. \(\vdash : P \in 1 \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \nu \in \text{NC induct}. \supset . P_{\nu } \in 1 \rightarrow 1 \quad[*121·34·341]\)

*121·35. \(\vdash : P \in 1 \rightarrow \text{Cls}. P_{\text{po}} \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct}. \supset . P_{\mu } \mid P_{\nu } = P_{\mu +_{c} \nu }\)

Dem. \[ \begin{array}{l} \vdash . *50·62. *121·302·322. &\supset \vdash : \text{Hp}. \supset . P_{\mu } \mid P_{0} = P_{\mu +_{c} 0} &\qquad \text{(1)}\\ \vdash . *121·332. &\supset \vdash \colon\ldotp \text{Hp}. \supset : \mu ,\nu \in \text{NC induct}. P_{\mu } \mid P_{\nu } = P_{\mu +_{c} \nu }. \supset .\\ &P_{\mu } \mid P_{\nu +_{c} 1} = P_{\mu +_{c} \nu } \mid P\\ [*121·332] &= P_{\mu +_{c} \nu +_{c} 1} &\qquad \text{(2)}\\ \vdash . (1).(2). *120·13.\supset \vdash . \text{Prop} \end{array} \]

*121·351. \(\vdash : P \in \text{Cls} \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct}. \supset . P_{\mu } \mid P_{\nu } = P_{\mu +_{c} \nu }\)

*121·352. \[\begin{align}\vdash : P \in (1 \rightarrow \text{Cls}) \cup (\text{Cls} \rightarrow 1). &P_{\text{po}} \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct}. \supset .\\ &P_{\mu } \mid P_{\nu } = P_{\nu } \mid P_{\mu } \quad[*121·35·351.*110·51]\end{align}\]

*121·36. \[\begin{align}\vdash : P \in (1 \rightarrow \text{Cls}) \cup (\text{Cls} \rightarrow 1). &P_{\text{po}} \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct} -\iota ʻ0.\supset .\\ &(P_{\mu })_{\nu } = P_{\mu \times _{c} \nu }\end{align}\]

Dem. \[ \begin{array}{l} \vdash . *121·321.\supset \vdash : \text{Hp}. &\supset . P_{\mu }\,\unicode{x2abd}\, P_{\text{po}}.\\ [*91·59·601] &\supset . (P_{\mu })_{\text{po}} \,\unicode{x2abd}\, J. &\qquad \text{(1)}\\ [*121·31·34·341] &\supset .(P_{\mu })_1 = P_{\mu } &\qquad \text{(2)}\\ \vdash . *121·332·333·352 . (1). \supset \\ \vdash \colon\ldotp \text{Hp}. \supset : (P_{\mu })_{\nu +_{c} 1} &= (P_{\mu })_{\nu } \mid P_{\mu } :\\ [*34·27] \supset : (P_{\mu })_{\nu } = P_{\mu \times _{c} \nu } . \supset . (P_{\mu })_{\nu +_{c} 1} &= P_{\mu \times _{c} \nu } \mid P_{\mu }\\ [*121·35·351] &= P_{(\mu \times _{c} \nu ) +_{c} \mu}\\ [*113·671] &= P_{\mu \times _{c} (\nu +_{c} 1)} &\qquad \text{(3)}\\ \vdash . (2). (3). *120·47. \supset \vdash . \text{Prop} \end{array} \]

[Pg 242]

*121·361. \[\begin{align}\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).&P_{\text{po}}\,\unicode{x2abd}\, J.\mu ,\nu \in \text{NC induct}-\iota ʻ0.\supset .\\ &(P_{\mu })_{\nu }=(P_{\nu })_{\mu } [*121·36.*113·27]\end{align}\]

*121·37. \(\vdash :P\in \text{Cls}\rightarrow 1.y\in P(x\vdash\dashv z).\supset .P(x\vdash\dashv z)=P(x\vdash\dashv y)\cup P(y\vdash\dashv z)\)

Dem. \[ \begin{array}{l} \vdash .*121·103.&\supset \vdash :\text{Hp}.\supset .xP_{*}y.yP_{*}z &\qquad \text{(1)}\\ \vdash .(1).*121·103.\supset \\ \vdash \colon\ldotp \text{Hp}.&\supset :w\in P(x\vdash\dashv z).\equiv .xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}z &\qquad \text{(2)}\\ \vdash .*96·302.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xP_{*}w.xP_{*}y.\supset :wP_{*}y.\lor.yP_{*}w &\qquad \text{(3)}\\ \vdash .(2).(3).*4·73.\supset \\ \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp w\in P(x\vdash\dashv z).\equiv :xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}z.wP_{*}y.\lor.\\ &xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}z.yP_{*}w &\qquad \text{(4)}\\ \vdash .*90·17.*4·73.\supset \vdash :wP_{*}y.yP_{*}z.&\equiv .wP_{*}z.wP_{*}y.yP_{*}z:\\ &yP_{*}w.wP_{*}z.\equiv .yP_{*}z.wP_{*}z.yP_{*}w &\qquad \text{(5)}\\ \vdash .(4).(5).\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp w\in P(x\vdash\dashv z).\equiv :xP_{*}w.xP_{*}y.yP_{*}z.wP_{*}y.\lor.\\ &xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}w:\\ [*90·17.*4·73] &\equiv:xP_{*}w.wP_{*}y.yP_{*}z.\lor.xP_{*}y.yP_{*}w.wP_{*}z:\\ [(1).*4·73] &\equiv :xP_{*}w.wP_{*}y.\lor.yP_{*}w.wP_{*}z:\\ [*121·103] &\equiv :w\in P(x\vdash\dashv y)\cup P(y\vdash\dashv z)\colon\colon \supset \vdash .\text{Prop} \end{array} \]

*121·371. \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x\vdash\dashv z).\supset .\\ &P(x\vdash\dashv z)=P(x\vdash\dashv y)\cup P(y\vdash\dashv z)=P(x \unicode{x27dd} y)\cup P(y\vdash\dashv z)\\ &=P(x\vdash\dashv y)\cup P(y\dashv z) \quad [\text{Proof as in *121·37}]\end{align}\]

*121·372. \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x\dashv z).\supset .\\ &P(x\dashv z)=P(x\dashv y)\cup P(y\dashv z)=P(x\dashv y)\cup P(y\vdash\dashv z)\end{align}\]

*121·373. \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x \unicode{x27dd} z).\supset .\\ &P(x \unicode{x27dd} z)=P(x \unicode{x27dd} y)\cup P(y \unicode{x27dd} z)= P(x\vdash\dashv y)\cup P(y \unicode{x27dd} z)\end{align}\]

*121·374. \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x\unicode{x2013}z).\supset .\\ &P(x\unicode{x2013}z)=P(x\dashv y)\cup P(y-z)=P(x\unicode{x2013}y)\cup P(y \unicode{x27dd} z)\\ &=P(x\dashv y)\cup P(y \unicode{x27dd} z)\end{align}\]

The proofs of these propositions are analogous to the proof of *121·37.

*121·38. \(\vdash :R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.\supset .R(x\vdash\dashv x)=\overleftarrow{R}_{*}ʻx \quad[*97·5]\)

*121·381. \(\vdash :R\in 1\rightarrow \text{Cls}.xR_{\text{po}}x.\supset .R(x\vdash\dashv x)=\overrightarrow{R}_{*}ʻx \quad[*97·501]\)

*121·382. \[\begin{align}\vdash :&R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.xR_{\text{po}}y.\supset .\\ &R(x\vdash\dashv x)=R(x\vdash\dashv y)=\overleftarrow{R}_{*}ʻx= R(y\vdash\dashv y) \quad[*97·5.*91·56]\end{align}\]

*121·383. \[\begin{align}\vdash :&R\in 1\rightarrow \text{Cls}.xR_{\text{po}}x.yR_{\text{po}}x.\supset .\\ &R(x\vdash\dashv x)=R(y\vdash\dashv x)=\overrightarrow{R}_{*}ʻx=R(y\vdash\dashv y)\end{align}\]

*121·384. \[\begin{align}\vdash :&R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).xR_{\text{po}}x.y\in R(x\vdash\dashv x).\supset .\\ &R(x\vdash\dashv x)= R(x\vdash\dashv y)=R(y\vdash\dashv x)=R(y\vdash\dashv y) \quad[*121·382·383]\end{align}\]

[Pg 243]

*121·39. \(\vdash \colon\ldotp R\in \text{Cls}\rightarrow 1.\supset :R(x\vdash\dashv y)\subset R(x\vdash\dashv z).\lor.R(x\vdash\dashv z)\subset R(x\vdash\dashv y)\)

Dem. \[ \begin{array}{l} \vdash .*96·302. &\supset \vdash \colon\ldotp \text{Hp}.xR_{*}y.xR_{*}z.\supset :yR_{*}z.\lor.zR_{*}y &\qquad \text{(1)}\\ \vdash .*121·37. &\supset \vdash :\text{Hp}.xR_{*}y.yR_{*}z.\supset .R(x\vdash\dashv y)\subset R(x\vdash\dashv z) &\qquad \text{(2)}\\ \vdash .*121·37. &\supset \vdash :\text{Hp}.xR_{*}z.zR_{*}y.\supset .R(x\vdash\dashv z)\subset R(x\vdash\dashv y) &\qquad \text{(3)}\\ \vdash .(1).(2).(3). &\supset \vdash \colon\ldotp \text{Hp}.xR_{*}y.xR_{*}z.\supset :\\ &R(x\vdash\dashv y)\subset R(x\vdash\dashv z).\lor.R(x\vdash\dashv z)\subset R(x\vdash\dashv y) &\qquad \text{(4)}\\ \vdash .*121·23. &\supset \vdash :{\sim}(xR_{*}y).\supset .R(x\vdash\dashv y)=\Lambda .\\ [*24·12] &\supset .R(x\vdash\dashv y)\subset R(x\vdash\dashv z) &\qquad \text{(5)}\\ \vdash .(5)\frac{z,\,y}{y,\,z}. &\supset \vdash :{\sim}(xR_{*}z).\supset .R(x\vdash\dashv z)\subset R(x\vdash\dashv y) &\qquad \text{(6)}\\ \vdash .(4).(5).(6).\supset \vdash .\text{Prop} \end{array} \]

The following series of propositions are concerned with proving *121·47, i.e. \[ R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).\supset .R(x\vdash\dashv z)\in \text{Cls induct}. \] The proof for \(R\in 1\rightarrow \text{Cls}\) follows from that for \(R\in \text{Cls}\rightarrow 1\) by *121·143. Confining ourselves, therefore, to \(R\in \text{Cls}\rightarrow 1\), we proceed as follows.

We prove first that, starting from z and going backwards, each new step adds only one term (which may not be distinct from all its predecessors); i.e. we have \[ R\in \text{Cls}\rightarrow 1.xRy.yR_{*}z.\supset .R(x\vdash\dashv z)=\iota ʻx\cup R(y\vdash\dashv z). \] From this it follows by induction that if \(R(z\vdash\dashv z)\) is an inductive class, so is \(R(x\vdash\dashv z)\). Thus we only have to prove that \(R(z\vdash\dashv z)\) is an inductive class. Here we must distinguish two cases, according as \({\sim}(zR_{\text{po}}z)\) or \(zR_{\text{po}}z\). In the former case, we have \[ \exists !R(z\vdash\dashv z).\supset .R(z\vdash\dashv z)=\iota ʻz, \] whence \(R(z\vdash\dashv z)\) is an inductive class, and therefore so is \(R(x\vdash\dashv z)\).

But in the latter case, when \(zR_{\text{po}}z\), the matter is more difficult. In this case, \(z\) is a member of a cycle, the cycle being \(R(z\vdash\dashv z)\). We have to prove that this cycle must be an inductive class. Given \(xR_{*}z\), \(x\) will be a member of this cycle if \(xR_{\text{po}}x\), and may be at the end of the tail of a \(Q\), if \({\sim}(xR_{\text{po}}x)\). (Cf. *96.)

By *96·453, we know that \(R\) is \(1\rightarrow 1\) when confined to \(R(z\vdash\dashv z)\). Hence[Pg 244] in \(R(z\vdash\dashv z)\), \(z\) has a unique predecessor, say \(a\). Assume \(a\neq z\). We then imagine a barrier placed between \(a\) and \(z\), i.e. we construct a relation \(S\) which is to hold between any two consecutive members of \(R(z\vdash\dashv z)\) except \(a\) and \(z\). Putting \(\alpha=R(z\vdash\dashv z)-\iota ʻa\), we have \(S=\alpha \upharpoonleft R\). Then the relation \(S\) generates an open series consisting of all the terms of \(R(z\vdash\dashv z)\); i.e. we have \[ {\sim}(aS_{\text{po}}a).S(z\vdash\dashv a)=R(z\vdash\dashv z). \] Hence, by our previous case, since \(S(z\vdash\dashv a)\) is an inductive class, so is \(R(z\vdash\dashv z)\).

If \(a = z\), then by *96·33 the cycle reduces to the single term \(z\), and therefore \(R(z\vdash\dashv z)\) is still an inductive class.

Hence \(R(z\vdash\dashv z)\), and therefore \(R(x\vdash\dashv z)\), is always an inductive class when \(R\in \text{Cls}\rightarrow 1\), which was to be proved.

*121·4. \(\vdash :R\in \text{Cls}\rightarrow 1.xRy.yR_{*}z.\supset .R(x\vdash\dashv z)=\iota ʻx\cup R(y\vdash\dashv z)\)

Dem. \[ \begin{array}{l} \vdash .*90·311.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xR_{*}w.&\equiv :x=w.\lor.xR\mid R_{*}w:\\ [*71·701.\text{Hp}] &\equiv :x=w.\lor.yR_{*}w &\qquad \text{(1)}\\ \vdash .*90·172.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x=w.\supset .wR_{*}z &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xR_{*}w.wR_{*}z.&\equiv :x=w.\lor.yR_{*}w.wR_{*}z &\qquad \text{(3)}\\ \vdash .(3).*121·103.\supset \vdash .\text{Prop} \end{array} \]

*121·41. \(\vdash :R\in \text{Cls}\rightarrow 1.R(z\vdash\dashv z)\in \text{Cls induct}.\supset .R(x\vdash\dashv z)\in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*121·4.*120·251.*90·172.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ &yR_{*}z.R(y\vdash\dashv z)\in \text{Cls induct}.xRy.\supset .xR_{*}z.R(x\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .(1).*90·112 \frac{\breve{R}}{R}.&\supset \vdash :\text{Hp}.xR_{*}z.\supset .R(x\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .*121·23.*120·212.&\supset \vdash :{\sim}(xR_{*}z).\supset .R(x\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(3)}\\ \vdash .(2).(3).\supset \vdash .\text{Prop} \end{array} \]

In virtue of this proposition, we have only to prove \(R(z\vdash\dashv z)\in \text{Cls induct}\). This is obvious when \({\sim}(zR_{\text{po}}z)\), for then either \(R(z\vdash\dashv z)=\iotaʻz\) or \(R(z\vdash\dashv z=\Lambda\). But when \(zR_{\text{po}}z\), it is more difficult.

*121·42. \(\vdash :R\in \text{Cls}\rightarrow 1.{\sim}(zR_{\text{po}}z).\supset .R(x\vdash\dashv z)\in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*121·303.\text{Transp}.*120·441.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻR(z\vdash\dashv z)\leq 1.\\ [*120·48] &\supset .\text{Nc}ʻR(z\vdash\dashv z)\in \text{NC induct}.\\ [*120·211] & \supset .R(z\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .(1).*121·41.\supset \vdash .\text{Prop} \end{array} \]

[Pg 245]

*121·43. \(\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.\supset .\text{E}!\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz)\)

Dem. \[ \begin{array}{l} \vdash .*91·52. \supset \vdash :\text{Hp}.&\supset .(\exists a).zR_{*}a.aRz &\qquad \text{(1)}\\ \vdash .*96·453. &\supset \vdash :\text{Hp}.\supset .(\overleftarrow{R}_{*}ʻz)\upharpoonleft R\in 1\rightarrow 1.\\ [*71·122] &\supset .\hat{a}(zR_{*}a.aRz)\in 1\cup \iota ʻ\Lambda &\qquad \text{(2)}\\ \vdash .(1).(2). \supset \vdash :\text{Hp}.&\supset .\hat{a}(zR_{*}a.aRz)\in 1.\\ [*52·15] &\supset .\text{E}!\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz):\supset \vdash .\text{Prop} \end{array} \]

*121·431. \[\begin{align}\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.a=\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz).&\alpha =\overleftarrow{R}_{*}ʻz-\iota ʻa.\\ &S=\alpha \upharpoonleft R.\supset .{\sim}(aS_{\text{po}}a)\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*35·61. \supset \vdash :\text{Hp}.&\supset .a{\sim}\in \text{D}ʻS.\\ [*91·504] &\supset .a{\sim}\in \text{D}ʻS_{\text{po}}.\\ [*33·14] &\supset .{\sim}(aS_{\text{po}}a):\supset \vdash .\text{Prop} \end{array} \]

*121·432. \(\vdash :\text{Hp}*121·431.\supset .S(z\vdash\dashv a)\in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*71·261.\supset \vdash :\text{Hp}.\supset .S\in \text{Cls}\rightarrow 1 &\qquad \text{(1)}\\ \vdash .(1).*121·431·42.\supset \vdash .\text{Prop} \end{array} \]

*121·433. \(\vdash :\text{Hp}*121·431.z\neq a.\supset .S(z\vdash\dashv a)=\overleftarrow{R}_{*}ʻz=R(z\vdash\dashv z)\)

Dem. \[ \begin{array}{l} \vdash .*96·11. \supset \vdash \colon\ldotp \text{Hp}.&\supset :zS_{*}w.\supset .zR_{*}w &\qquad \text{(1)}\\ \vdash .*51·3.*91·504. \supset \vdash \colon\ldotp \text{Hp}.&\supset :z\in a.z\in \text{D}ʻR:\\ [*35·61] &\supset :z\in \text{D}ʻS:\\ [*90·12] &\supset :zS_{*}z &\qquad \text{(2)}\\ \vdash .(1).*90·16. \supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp zS_{*}w.wRy.\supset :w\in \alpha \cup \iota ʻa.wRy:\\ [*35·1] &\supset :wSy.\lor.w=a.wRy:\\ [\text{Hp}.*71·171] &\supset :wSy.\lor.y=z:\\ [*90·16·17.(2)] &\supset :zS_{*}y &\qquad \text{(3)}\\ \vdash .(2).(3).*90·112. \supset \vdash \colon\ldotp \text{Hp}.&\supset :zR_{*}w.\supset .zS_{*}w: &\qquad \text{(4)}\\ [\text{Hp}] &\supset \vdash :zS_{*}a &\qquad \text{(5)}\\ \vdash .*71·171. \supset \vdash :\text{Hp}.&aRy.\supset .y=z &\qquad \text{(6)}\\ \vdash .*91·542·504.*35·61. &\supset \vdash \colon\ldotp \text{Hp}.\supset :wS_{*}a.w\neq a.wRy.\supset .wS_{\text{po}}a.wSy.\\ [*92·111] &\supset .yS_{*}a &\qquad \text{(7)}\\ \vdash .(5).(6).(7). &\supset \vdash \colon\ldotp \text{Hp}.\supset :wS_{*}a.wRy.\supset .yS_{*}a &\qquad \text{(8)}\\ \vdash .(5).(8).*90·112. &\supset \vdash \colon\ldotp \text{Hp}.\supset :zR_{*}y.\supset .yS_{*}a &\qquad \text{(9)}\\ \vdash .(4).(9). &\supset \vdash \colon\ldotp \text{Hp}.\supset :zR_{*}y.\supset .zS_{*}y.yS_{*}a &\qquad \text{(10)}\\ \vdash .(1) \frac{y}{w}.(10). &\supset \vdash \colon\ldotp \text{Hp}.\supset :zS_{*}y.yS_{*}a.\equiv .zR_{*}y:\\ [*121·103] \supset :S(z\vdash\dashv a)&=\overleftarrow{R}_{*}ʻz\\ [*121·38] &=R(z\vdash\dashv z)\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

[Pg 246]

*121·434. \(\vdash :\text{Hp}*121·431.z=a.\supset .\overleftarrow{R}_{*}ʻz=R(z\vdash\dashv z)=\iota ʻz\)

Dem. \[ \begin{array}{l} \vdash .*32·18.&\supset \vdash :\text{Hp}.\supset .zRz.\\ [*96·33] &\supset .\overleftarrow{R}_{*}ʻz=\iota ʻz. &\qquad \text{(1)}\\ [*121·38] &\supset .R(z\vdash\dashv z)=\iota ʻz &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·44. \(\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.\supset .R(z\vdash\dashv z)\in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*121·43·432·433.\supset \\ \vdash :\text{Hp}.z\neq \breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz).\supset .R(z\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .*121·434.*120·213.\supset \\ \vdash :\text{Hp}.z=\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz).\supset .R(z\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·441. \(\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·44·41]\)

*121·45. \(\vdash :R\in \text{Cls}\rightarrow 1.\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·42·441]\)

*121·46. \(\vdash :R\in 1\rightarrow \text{Cls}.\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·45·143]\)

*121·47. \(\vdash :R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·45·46]\)

*121·48. \[\begin{align}\vdash \colon\ldotp &R\in \text{Cls}\rightarrow 1.\supset :\\ &\text{Nc}ʻR(x\vdash\dashv y)<\text{Nc}ʻR(x\vdash\dashv z).\equiv .\exists !R(x\vdash\dashv z)-R(x\vdash\dashv y)\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·39.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !R(x\vdash\dashv z)-R(x\vdash\dashv y).\equiv .\\ &R(x\vdash\dashv y)\subset R(x\vdash\dashv z).R(x\vdash\dashv y)\neq R(x\vdash\dashv z).\\ [*120·7.*121·45] &\supset .\text{Nc}ʻR(x\vdash\dashv y)<\text{Nc}ʻR(x\vdash\dashv z) &\qquad \text{(1)}\\ \vdash .*117·222·29.&\supset \vdash :\text{Nc}ʻR(x\vdash\dashv y)<\text{Nc}ʻR(x\vdash\dashv z).\supset .\\ &{\sim}\{R(x\vdash\dashv z)\subset R(x\vdash\dashv y)\}.\\ [*24·55] &\supset .\exists !R(x\vdash\dashv z)-R(x\vdash\dashv y) &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·481. \[\begin{align}\vdash \colon\ldotp R\in \text{Cls}\rightarrow 1.\supset :\text{Nc}ʻR(x\vdash\dashv y)\leq \text{Nc}ʻ&R(x\vdash\dashv z).\equiv .\\ &R(x\vdash\dashv y)\subset R(x\vdash\dashv z)\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·45.*120·441.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\text{Nc}ʻR(x\vdash\dashv y)\leq \text{Nc}ʻR(x\vdash\dashv z).&\equiv .{\sim}\{\text{Nc}ʻR(x\vdash\dashv z)<\text{Nc}ʻR(x\vdash\dashv y)\}.\\ [*121·48] &\equiv .{\sim}\exists !R(x\vdash\dashv y)-R(x\vdash\dashv z).\\ [*24·55] &\equiv .R(x\vdash\dashv y)\subset R(x\vdash\dashv z)\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

The above proposition is used in the proof of *122·35, which is an important proposition in the theory of progressions.

[Pg 247]

The following propositions are concerned with the identification of such relations as \(P_{\nu }\) with powers of \(P\) in the sense of *91.

*121·5. \[\begin{align}\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).&P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\\ &\text{finid}ʻP=\text{Potid}ʻP.\text{fin}ʻP=\text{Pot}ʻP\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·302·31. \supset \vdash :\text{Hp}.&\supset .P_{0}=I\upharpoonright CʻP.P_{1}=P &\qquad \text{(1)}\\ \vdash .(1).*121·332·333·352.&\supset \vdash \colon\ldotp \text{Hp}.\nu \in \text{NC induct}.\supset :P_{\nu +_{c}1}=P_{\nu }\mid P: &\qquad \text{(2)}\\ [*91·341] &\supset :P_{\nu }\in \text{Potid}ʻP.\supset .P_{\nu +_{c}1}\in \text{Potid}ʻP:P_{\nu }\in \text{Pot}ʻP.\supset .P_{\nu +_{c}1}\in \text{Pot}ʻP &\qquad \text{(3)}\\ \vdash .(1).*91·35.&\supset \vdash :\text{Hp}.\supset .P_{0}\in \text{Potid}ʻP.P_{1}\in \text{Pot}ʻP &\qquad \text{(4)}\\ \vdash .(3).(4).*120·13·47.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset .P_{\nu }\in \text{Potid}ʻP:\\ &\nu \in \text{NC induct}-\iota ʻ0.\supset .P_{\nu }\in \text{Pot}ʻP:\\ [*121·12·121] &\supset :\text{finid}ʻP\subset \text{Potid}ʻP.\text{fin}ʻP\subset \text{Pot}ʻP &\qquad \text{(5)}\\ \vdash .(2).*121·121.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset .P_{\nu }\mid P\in \text{fin}ʻP:\\ [*121·12] & \supset :Q\in \text{finid}ʻP.\supset .Q\mid P\in \text{fin}ʻP:\\ [(1).*91·17·171] & \supset :\text{Potid}ʻP\subset \text{finid}ʻP.\text{Pot}ʻP\subset \text{fin}ʻP &\qquad \text{(6)}\\ \vdash .(5).(6).\supset \vdash .\text{Prop} \end{array} \]

*121·501. \[\begin{align}\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, &J.\dot{\exists} !P.\supset .\\ &\text{Pot}ʻP=\text{finid}ʻP-\iota ʻP_{0}=\text{fin}ʻP\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·302.\supset \vdash :\text{Hp}.\supset .\dot{\exists} !P_{0} &\qquad \text{(1)}\\ \vdash .(1).*121·5·327.\supset \vdash .\text{Prop} \end{array} \]

*121·502. \[\begin{align}\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}&\,\unicode{x2abd}\, J.\supset .\\ &\dot{s} ʻ(\text{finid}ʻP-\iota ʻP_{0})=P_{\text{po}}=\dot{s} ʻ\text{fin}ʻP\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*91·504.*33·24.*121·5.\supset \vdash :P=\dot{\Lambda} .\supset .\dot{s} ʻ(\text{finid}ʻP-\iota ʻP_{0})=\dot{\Lambda} =P_{\text{po}} &\qquad \text{(1)}\\ \vdash .(1).*121·501·5.\supset \vdash .\text{Prop} \end{array} \]

*121·51. \(\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{2}=P^{2}.P_{3}=P^{3}.\text{etc}.\)

Dem. \[ \begin{array}{l} \vdash .*121·31. \supset \vdash :\text{Hp}.&\supset .P_{1}=P &\qquad \text{(1)}\\ \vdash .*121·332·333. \supset \vdash :\text{Hp}.\supset .P_{2}&=P_{1}\mid P_{1}\\ [(1)] & =P^{2} &\qquad \text{(2)}\\ \vdash .*121·332·333·352. \supset \vdash :\text{Hp}.\supset .P_{3}&=P_{2}\mid P_{1}\\ [(1).(2)] &=P^{3} &\qquad \text{(3)}\\ \vdash .(2).(3). \text{etc.} .\supset \vdash .\text{Prop} \end{array} \]

*121·52. \[\begin{align}&\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\dot{s} ʻ\text{finid}ʻP=P_{*}\\ &[*121·5.*91·55]\end{align}\]

We shall at a later stage (*301) give a general definition of \(P^\nu\). When this definition has been introduced, we shall be able to prove, with the hypothesis of *121·51, \[ \nu \in \text{NC induct}.\supset .P_{\nu }=P^\nu . \] The definition of \(P^\nu\) is postponed on account of various complications which render a general definition of \(P^\nu\) difficult. The chief difficulty arises when[Pg 248] \(\dot{\exists} !P\dot{\cap} I\). Thus suppose we have \(yPy\); we shall also have \(yP^{2}y\), \(yP^{3}y\), etc. Hence if we have \(xPy\), we have \[ \nu \in \text{NC induct}-\iota ʻ0.\supset _{\nu }.xP^\nu y. \] Again, suppose this case excluded, but suppose \[ (\exists \mu ,y).\mu \in \text{NC induct}.y\in P(x\vdash\dashv z).yP^\mu y . \] Then we shall have \[ \nu \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda .\supset .yP^{\mu \times _{c}\nu }y. \] Thus the general definition of \(P^\nu\) has to be complicated, except when \(P_{\text{po}}\,\unicode{x2abd}\, J\).

The following propositions are concerned with the series of relations \(P_{\nu }\) and the series of terms \(\nu _{P}\). The relation \(P_{\nu }\) holds between two terms (roughly speaking) when it requires \(\nu\) steps to get from the first to the second; the term \(\nu _{P}\) is the \(\nu\)th term starting from \(BʻP\), which, when it exists, is \(1_{P}\). In order that \(\nu _{P}\) should exist, it is necessary that \(BʻP\) should exist, and that there should be just one term \(x\) in the field of \(P\) such that the interval from \(BʻP\) to \(x\) (both included) consists of \(\nu\) terms. When this is the case for all inductive cardinals from 1 to \(\nu\), we can say that \(P\) generates a series starting from \(BʻP\) and having at least \(\nu\) terms, each correlated with one of the cardinals in the interval from 1 to \(\nu\), both included; i.e. the series has a \(\mu\)th term, whenever \(1\leq \mu \leq \nu\). If this holds for all inductive values of \(\nu\), the family of \(BʻP\) is a progression[9]. (It will be observed that all such terms as \(\nu _{P}\) belong to the family of \(BʻP\), which need not form the whole field of \(P\).)

*121·6. \(\vdash \colon\ldotp \nu \neq 0.\supset .f(\nu _{P}).\equiv .f[\breve{\iota} ʻ\hat{y} \{\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv y)=\nu\}]\)

Dem. \[ \begin{array}{l} \vdash .*121·11.*120·414·416.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ f[\breve{\iota} ʻ\hat{y} \{\text{Nc}ʻP (BʻP \vdash\dashv y) = \nu\}] . &\equiv . f[\breve{\iota} ʻ\hat{y} \{(BʻP)P_{\nu -_{c}1}y\}].\\ [*121·13] &\equiv.f(\nu _{P}):\supset \vdash .\text{Prop} \end{array} \]

*121·601. \(\vdash :\text{E}!BʻP.\supset .BʻP=1_{P}.{\sim}{(BʻP)P_{\text{po}}(BʻP)}\)

Dem. \[ \begin{array}{l} \vdash .*91·504.*93·1.&\supset \vdash .{\sim}\{(BʻP)P_{\text{po}}(BʻP)\}. &\qquad \text{(1)}\\ [*121·301] \supset \vdash \colon\ldotp \text{E}!BʻP.&\supset :(BʻP)P_{0}y.\equiv _{y}.BʻP=y:\\ [*31·17] &\supset :BʻP=\breve{P} _{0}ʻBʻP:\\ [*121·13] &\supset :BʻP=1_{P} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·602. \(\vdash :\text{E}!BʻP.P\in 1\rightarrow 1.\supset \breve{P} ʻBʻP=2_{P}\)

Dem. \[ \begin{array}{l} \vdash .*121·306·601.&\supset \vdash :\text{Hp}.\supset .P(BʻP\vdash\dashv \breve{P} ʻBʻP)\in 2 &\qquad \text{(1)}\\ \vdash .*121·23·601.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp (BʻP)P_{\text{po}}y.\supset .BʻP,y\in P(BʻP\vdash\dashv y).BʻP\neq y\colon\ldotp \\ [*54·53.*121·303] \supset \colon\ldotp P(BʻP\vdash\dashv y)\in 2.&\supset :P(BʻP\vdash\dashv y)=\iota ʻBʻP\cup \iota ʻy.(BʻP)P_{\text{po}}y:\\ [*92·111] &\supset :(\breve{P} ʻBʻP)P_{*}y.P(BʻP\vdash\dashv y)=\iota ʻBʻP\cup \iota ʻy:\\ [*121·103·601] &\supset :\breve{P} ʻBʻP\in \iota ʻBʻP\cup \iota ʻy.\breve{P} ʻBʻP\neq BʻP:\\ [*51·232] &\supset :y=\breve{P} ʻBʻP &\qquad \text{(2)}\\ \vdash .(1).(2).*121·6.\supset \vdash .\text{Prop} \end{array} \]

[Pg 249]

*121·61. \[\begin{align}\vdash :P\in 1\rightarrow \text{Cls}.P_{\text{po}}\,\unicode{x2abd}\, J.x\in &sʻ\text{gen}ʻP.\supset .\\ &(\exists a,\nu ).a BP.\nu \in \text{NC induct}.a P_{\nu }x\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*93·36. &\supset \vdash \colon\ldotp P\in 1\rightarrow \text{Cls}.x\in sʻ\text{gen}ʻP.\supset .(\exists a).a BP.a P_{*}x &\qquad \text{(1)}\\ \vdash .*121·52. &\supset \vdash \colon\ldotp P\in 1\rightarrow \text{Cls}.P_{\text{po}}\,\unicode{x2abd}\, J.\supset :a P_{*}x.\equiv .a(\dot{s} ʻ\text{finid}ʻP)x &\qquad \text{(2)}\\ \vdash .(1).(2).&\supset \vdash :\text{Hp}.\supset .(\exists a).\alpha BP.a (\dot{s} ʻ\text{finid}ʻP)x.\\ [*121·12] &\supset .(\exists a,\nu ).a BP.\nu \in \text{NC induct}.a P_{\nu }x:\supset \vdash .\text{Prop} \end{array} \]

*121·62. \[\begin{align}\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.(BʻP)&P_{*}x.\supset .\\ &(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ0.x=\nu _{P}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·52. \supset \vdash :\text{Hp}.&\supset .(BʻP)(\dot{s} ʻ\text{finid}ʻP)x.\\ [*121·12] &\supset .(\exists \nu ).\nu \in \text{NC induct}.(BʻP)P_{\nu }x &\qquad \text{(1)}\\ \vdash .*121·341.&\supset \vdash :\text{Hp}.\nu \in \text{NC induct}.\supset .P_{\nu }\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ \vdash .(1).(2). \supset \vdash :\text{Hp}.&\supset .(\exists \nu ).\nu \in \text{NC induct}.x=\breve{P} _{\nu }ʻBʻP.\\ [*121·13] &\supset .(\exists \nu ).\nu \in \text{NC induct}.x=(\nu +_{c}1)_P.\\ [*120·471] &\supset .(\exists \mu ).\mu \in \text{NC induct}-\iota ʻ0.x=\mu _{P}:\supset \vdash .\text{Prop} \end{array} \]

*121·63. \(\vdash :\text{E}!\nu _{P}.\supset .\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv \nu _{P})=\nu\)

Dem. \[ \begin{array}{l} \vdash .*121·13·131.\supset \vdash :\text{Hp}.&\supset .(BʻP)P_{\nu -_{c}1}\nu _{P}.\\ [*121·11] &\supset .\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv \nu _{P})=\nu :\supset \vdash .\text{Prop} \end{array} \]

*121·631. \[\begin{align}\vdash \colon\ldotp P\in \text{Cls}\rightarrow 1.&P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.\supset :\\ &\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv y)=\nu .\equiv .y=\nu _{P}.\equiv .(BʻP)P_{\nu -_{c}1}y\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*120·414·416.*121·11.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv y)=\nu .&\equiv .(BʻP)P_{\nu -{c}1}y. &\qquad \text{(1)}\\ [*121·341] &\equiv .y=\breve{P} _{\nu -_{c}1}ʻBʻP.\\ [*121·13] &\equiv .y=\nu _{P} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·632·633 are required for proving *121·634.

[Pg 250]

*121·632. \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.y=\nu _{P}.yPz.\supset .z=(\nu +_{c}1)_{P}\)

Dem. \[ \begin{array}{l} \vdash .*121·13. \supset \vdash :\text{Hp}.&\supset .(BʻP)P_{\nu -_{c}1}y.yPz.\\ [*121·333·352] &\supset .(BʻP)P_{\nu }z.\\ [*121·631] &\supset .z=(\nu +_{c}1)_{P}:\supset \vdash .\text{Prop} \end{array} \]

*121·633. \[\begin{align}\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in &\text{NC induct}-\iota ʻ0.\nu _{P}\in \text{D}ʻP.\supset .\\ &\text{E}!(\nu +_{c}1)_{P}.(\nu +_{c}1)_{P}=\breve{P} ʻ\nu _{P}\\ &[*121·632]\end{align}\]

*121·634. \[\begin{align}&\vdash \colon\ldotp P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.\supset :\nu _{P}\in \text{D}ʻP.\equiv .\text{E}!(\nu +_{c}1)_{P}\\ &[*121·633·631·333·352]\end{align}\]

*121·635. \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\text{E}!\nu _{P}.\supset .\nu \in \text{NC induct}-\iota ʻ0\)

Dem. \[ \begin{array}{l} \vdash .*121·63·45. &\supset \vdash :\text{Hp}.\supset .\nu \in \text{NC induct} &\qquad \text{(1)}\\ \vdash .*121·13. \supset \vdash :\text{E}!\nu _{P}.&\supset .\dot{\exists} !P_{(\nu -_{c}1)}.\\ [*121·272] &\supset .(\nu -_{c}1)+_{c}1>0.\\ [*120·416] &\supset .\nu >0 &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·636. \[\begin{align}\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.&\text{E}!\nu _{P}.{\sim}\text{E}!(\nu +_{c}1)_{P}.\supset .\\ &\overleftarrow{P}_{*}ʻBʻP=P(BʻP\vdash\dashv \nu _{P}).\text{N}_{0}\text{c}ʻ\overleftarrow{P}_{*}ʻBʻP=\nu\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·635. \supset \vdash :\text{Hp}.&\supset .\nu \in \text{NC induct}-\iota ʻ0. &\qquad \text{(1)}\\ [*121·634.\text{Hp}] &\supset .\nu _{P}{\sim}\in \text{D}ʻP &\qquad \text{(2)}\\ \vdash .(1).*121·63. \supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp \exists !P(BʻP\vdash\dashv \nu _{P})\colon\ldotp \\ [*121·23] &\supset \colon\ldotp (BʻP)P_{*}\nu _{P}\colon\ldotp \\ [*96·302.*91·542] &\supset \colon\ldotp (BʻP)P_{*}z.\supset :zP_{*}\nu _{P}.\lor.\nu _{P}P_{\text{po}}z &\qquad \text{(3)}\\ \vdash .(2).(3).*91·504. \supset \vdash \colon\ldotp \text{Hp}.&\supset :(BʻP)P_{*}z.\supset .zP_{*}\nu _{P}:\\ [*4·71] &\supset :(BʻP)P_{*}z.\equiv .(BʻP)P_{*}z.zP_{*}v_{P}:\\ [*121·103] &\supset :\overleftarrow{P}_{*}ʻBʻP=P(BʻP\vdash\dashv \nu _{P}): &\qquad \text{(4)}\\ [*121·63] &\supset :\text{N}_{0}\text{c}ʻ\overleftarrow{P}_{*}ʻBʻP=\nu &\qquad \text{(5)}\\ \vdash .(4).(5).\supset \vdash .\text{Prop} \end{array} \]

*121·637. \(\vdash :\text{E}!\nu _{P}.\supset .\nu _{P}\in CʻP\)

Dem. \[ \begin{array}{l} \vdash .*121·13.*14·28. \supset \vdash :\text{E}!\nu _{P}.&\equiv .\nu _{P}.=\breve{P} _{\nu -_{c}1}ʻBʻP.\\ [*121·322] &\supset .\nu _{P}\in CʻP:\supset \vdash .\text{Prop} \end{array} \]

[Pg 251]

*121·638. \(\vdash \colon\ldotp \text{E}!(\nu +_{c}1)_{P}.\supset :(BʻP)P_{\nu }x.\equiv .x=(\nu +_{c}1)_{P}:(\nu +_{c}1)-_{c}1=\nu \)

Dem. \[ \begin{array}{l} \vdash .*121·13.\supset \vdash :\text{E}!(\nu +_{c}1)_{P}.&\equiv .\text{E}!\breve{P}_\{(\nu +_{c}1)\}-_{c}1ʻBʻP. &\qquad \text{(1)}\\ [*121·272] &\supset .(\nu +_{c}1)-_{c}1\geq 0.\\ [*14·21] &\supset .\text{E}!(\nu +_{c}1)-_{c} 1.\\ [*14·22.(*120·411)] &\supset .(\nu +_{c}1)-_{c}1=\nu &\qquad \text{(2)}\\ \vdash .(2).\supset \vdash \colon\ldotp \text{Hp}.\supset :(BʻP)P_{\nu }x.&\equiv .(BʻP)P_{(\nu +_{c}1)-_{c}1}x.\\ [(1).*30·4] &\equiv .x=\breve{P} _{(\nu +_{c}1)-_{c}1}ʻBʻP.\\ [*121·13] &\equiv .x=(\nu +_{c}1)_{P} &\qquad \text{(3)}\\ \vdash .(3).(2).\supset \vdash . \text{Prop} \end{array} \]

*121*64. \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\supset .\text{E}!\nu _{P}\)

Dem. \[ \begin{array}{l} \vdash .*121·636.&\supset \vdash \colon\ldotp \text{Hp}.\text{E}!\nu _{P}.\supset :{\sim}\text{E}!(\nu +_{c} 1)_{P}.\supset .\text{N}_{0}\text{c}ʻ\overleftarrow{P}_{*}ʻBʻP=\nu &\qquad \text{(1)}\\ \vdash .*120·428.\supset \vdash :\nu \in \text{NC induct}.\exists !\nu +_{c}1.&\supset .\nu +_{c}1>\nu .\\ [*117·281] &\supset .{\sim}(\nu \geq \nu +_{c}1) &\qquad \text{(2)}\\ \vdash .*117·15. &\supset \vdash ·{\sim} \exists !\nu +_{c}1.\supset .{\sim}(\nu \geq \nu +_{c}1) &\qquad \text{(3)}\\ \vdash .(2).(3). &\supset \vdash :\nu \in \text{NC induct}.\supset .{\sim}(\nu \geq \nu +_{c}1)&\qquad \text{(4)}\\ \vdash .(1).(4). &\supset \vdash \colon\ldotp \text{Hp}.\text{E}!\nu _{P}.\supset :\\ &{\sim}\text{E}!(\nu +_{c}1)_{P}.\supset .{\sim}(\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu +_{c}1):\\ [\text{Transp}] &\supset :\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu +_{c}1.\supset .\text{E}!(\nu +_{c}1)_{P} &\qquad \text{(5)}\\ \vdash .(5).\text{Syll}.*117·6.\supset \vdash \colon\ldotp \text{Hp}:&\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\supset .\text{E}!\nu _{p}:\supset :\\ &\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu +_{c}1.\supset .\text{E}!(\nu +_{c}1)_{P} &\qquad \text{(6)}\\ \vdash .*14·21.*121·601.&\supset \vdash :\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq 1.\supset .\text{E}!1_{P} &\qquad \text{(7)}\\ \vdash .(6).(7).*120·473.\supset \\ \vdash \colon\ldotp \text{Hp}.\supset :\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\supset .\text{E}!\nu _{P}\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*121·641. \[\begin{align}\vdash \colon\ldotp P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in &\text{NC induct}-\iota ʻ0.\supset :\\ &\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\equiv .\text{E}!\nu _{P}\\ &[*121·64·63·32]\end{align}\]

*121·65. \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\mu \neq 0.\text{E}!(\mu +_{c}\nu )_{P}.\supset .\mu _{P}P_{\nu }(\mu +_{c}\nu )_{P}\)

Dem. \[ \begin{array}{l} \vdash .*121·631·635·64.*120·452.\supset \\ \vdash :\text{Hp}.&\supset .(BʻP)P_{\mu -_{c}1}\mu _{P}.(BʻP)P_{\mu +_{c}\nu -_{c}1}(\mu +_{c}\nu )_{P}.\\ [*121·351.*120·424]&\supset .(BʻP)P_{\mu -_{c}1}\mu _{P}.(BʻP)(P_{\mu -_{c}1}\mid P_{\nu })(\mu +_{c}\nu )_{P}.\\ [*121·341.*72·591] &\supset .\mu _{P}P_{\nu }(\mu +_{c}\nu )_{P}:\supset \vdash .\text{Prop} \end{array} \]

[Pg 252]

*121·66. \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\text{Nc}ʻP(BʻP\vdash\dashv x)>\nu .\supset .x\in \text{ᗡ}ʻP_{\nu }\)

Dem. \[ \begin{array}{l} \vdash .*121·45.*120·48.\supset \vdash :\text{Hp}.&\supset .\nu \in \text{NC induct}.\\ [*120·429] &\supset .\text{Nc}ʻP(BʻP\vdash\dashv x)\geq \nu +_{c}1.\\ [*117·31] &\supset .(\exists \mu ).\text{Nc}ʻP(BʻP\vdash\dashv x)=\nu +_{c}1+_{c}\mu .\\ [*121·11] &\supset .(\exists \mu ).(BʻP)P_{\nu +_{c}\mu }x.\\ [*121·351·352] &\supset .(\exists \mu ).(BʻP)(P_{\mu }\mid P_{\nu })x.\\ [*34·36] &\supset .x\in \text{ᗡ}ʻP_{\nu }:\supset \vdash .\text{Prop} \end{array} \]

The following proposition is used in *122·38·381.

*121·7. \(\vdash :R\in 1\rightarrow 1.aBR.aR_{*}x.\supset .\overrightarrow{R}_{*}ʻx=R(a\vdash\dashv x).\overrightarrow{R}_{*}ʻx\in \text{Cls induct}\)

Dem. \[ \begin{array}{l} \vdash .*96·25.\supset \vdash \colon\ldotp \text{Hp}.&\supset :yR_{*}x.\supset .aR_{*}y:\\ [*4·71] &\supset :yR_{*}x.\equiv .aR_{*}y.yR_{*}x:\\ [*121·103] &\supset :\overrightarrow{R}_{*}ʻx=R(a\vdash\dashv x) &\qquad \text{(1)}\\ \vdash .(1).*121·45.\supset \vdash .\text{Prop} \end{array} \]

*121·71. \[\begin{align}\vdash \colon\ldotp R\in 1\rightarrow 1:x\in sʻ\text{gen}ʻR.\lor.(\exists y).y\in &\overleftrightarrow{R}_{*}ʻx.yR_{\text{po}}y:\supset .\\ &\overrightarrow{R}_{*}ʻx\in \text{Cls induct}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*121·7.*93·36.&\supset \vdash :R\in 1\rightarrow 1.x\in sʻ\text{gen}ʻR.\supset .\overrightarrow{R}_{*}ʻx\in \text{Cls induct} &\qquad \text{(1)}\\ \vdash .*97·55·111.\supset \vdash \colon\ldotp R\in 1\rightarrow 1:(\exists y).&y\in \overleftrightarrow{R}_{*}ʻx.yR_{\text{po}}y:\supset :\\ &y\in \overleftrightarrow{R}_{*}ʻx .\supset _{y}.yR_{\text{po}}y:x\in \overleftrightarrow{R}_{*}ʻx:\\ [*10·26] &\supset :xR_{\text{po}}x:\\ [*121·381] &\supset :\overrightarrow{R}_{*}ʻx=R(x\vdash\dashv x):\\ [*121·45] &\supset :\overrightarrow{R}_{*}ʻx\in \text{Cls induct} &\qquad \text{(2)}\\ \vdash .(1).(2).\supset \vdash .\text{Prop} \end{array} \]

*121·72. \[\begin{align}&\vdash :R\in 1\rightarrow 1.\overrightarrow{R}_{*}ʻx{\sim}\in \text{Cls induct}.\supset .x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR.R_{\text{po}}\unicode{x0294f}\overleftrightarrow{R}_{*}ʻx\,\unicode{x2abd}\, J\\ &[*121·71.\text{Transp}.*93·271.*120·212.*50·24]\end{align}\]

Summary of *122.

By a "progression" we mean a series which is like the series of the inductive cardinals in order of magnitude (assuming that all inductive cardinals exist), i.e. a series whose terms can be called \[ 1_{R}, 2_{R}, 3_{R}, ... \nu _{R}, ..., \] where every term of the series is correlated with some inductive cardinal, and every inductive cardinal is correlated with some term of the series. Such series belong to the relation-number (cf. *152 and *263) which Cantor calls \(\omega\). Their generating relation may be taken to be the transitive relation of earlier and later, or the one-one relation of immediate predecessor to immediate successor. We shall reserve the notation \(\omega\) for the transitive generating relations of progressions; for the present, we are concerned with the one-one relations which generate progressions. The class of these relations we shall call "\(\text{Prog}\)."

It is not convenient to define a progression as a series which is ordinally similar to that of the inductive cardinals, both because this definition only applies if we assume the axiom of infinity, and because we have in any case to show that (assuming the axiom of infinity) the series of inductive cardinals has certain properties, which can be used to afford a direct definition of progressions. The existence of progressions, however, is only obtainable by means of the axiom of infinity, and is then most easily obtained from the fact that the inductive cardinals form a progression. We shall not consider the existence-theorem until the next number (*123).

From this number onwards convention \(\text{Infin}\,T\) of the Prefatory Statement is used when relevant.

The characteristics of the generating relation \(R\) of a progression, which we employ in the definition, are the following:

(1) \(R\) is a one-one relation;

(2) there is a first term, i.e. \(\text{E}!BʻR\);

(3) the whole field is contained in the posterity of the first term, i.e. \(CʻR=\overleftarrow{R}_{*}ʻBʻR\). (If this failed, \(CʻR\) would consist of two or more distinct families, of which, since we have \(\text{E}!BʻR\), all but one would have to be cyclic families.)

(4) every term of the field has a successor, i.e. the series is endless. This is secured \(\text{ᗡ}ʻR\subset \text{D}ʻR\), or (what is equivalent) \(CʻR=\text{D}ʻR\).

[Pg 254]

These four properties suffice to define the one-one generating relations of progressions. It will be observed that (2), (3) and (4) are all secured by \[ \text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR. \]

This secures \(\text{E}!BʻR\), by *14·21; it secures \(\text{ᗡ}ʻR\subset \text{D}ʻR\), by *37·25 and *90·163; hence, by *33·181, \(\text{D}ʻR=CʻR\), and therefore \[ CʻR=\overleftarrow{R}_{*}ʻBʻR. \]

Hence our definition of progressions is \[ \text{Prog}=(1\rightarrow 1)\cap \hat{R} (\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR) \quad\text{Df}. \]

Instead of stating in the definition that \(R\) is to be a one-one relation, it is sufficient to put \(R\in\text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J\), which, with \(\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR\), implies \(R\in 1\rightarrow 1\), and may be substituted for \(R\in 1\rightarrow 1\) without altering the force of the definition (*122·17).

In the present number we shall prove, among other propositions, that every existent class contained in a progression has a first term (*122·23), i.e. that progressions are well-ordered series; that in a progression \(R_{\text{po}}\,\unicode{x2abd}\, J\) (*122·16), which makes the propositions of *121 available; that if \(\nu\) is any inductive cardinal other than 0, \(\nu _{R}\) exists (*122·33), i.e. the series has a \(\nu\)th term; that any class contained in \(\text{D}ʻR\) and having a last term is an inductive class (*122·43), and that any class contained in \(\text{D}ʻR\) and not having a last term is itself the domain of a progression (*122·45), so that every class contained in \(\text{D}ʻR\) is either inductive or the domain of a progression (*122·46); that if \(P\) is a many-one, and \(x\) a member of its domain, and if the descendants of \(x\) have no last term and are none of them descendants of themselves, then \(P\) arranges these descendants in a progression (*122·51); and that the same holds if \(P\) is a one-one and \({\sim}(xPx)\) (*122·52); and that if \(P\in 1\rightarrow 1\) and \(x\) belongs to one of the generations of \(P\), but not to one of the generations of \(\breve{P}\), then \(P\) arranges the whole family of \(x\) in a progression (*122·54).

The following general observations on the families of one-one relations may serve to elucidate the bearing of the propositions of this section.

Given any relation \(P\), we call \(\overleftrightarrow{P}_{*}ʻx\), i.e. \(\overrightarrow{P}_{*}ʻx\cup \overleftarrow{P}_{*}ʻx\) the family of \(x\). If \(P\) is a one-one, this family may be of four different kinds. (1) It may be a closed series, like the angles of a polygon. This occurs if \(xP_{\text{po}}x\). In this case the family forms an inductive class. (2) It may be an open series with a beginning and an end; this occurs if \[ {\sim}(xP_{\text{po}}x).\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx. \]

In this case also the family forms an inductive class. (3) It may be an[Pg 255] open series with a beginning and no end, or an end and no beginning. This occurs if \[ {\sim}(xP_{\text{po}}x).\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.{\sim}\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx, \] or if \[ {\sim}(xP_{\text{po}}x).{\sim}\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx. \] In this case, the series is of the type \(\omega\) or \(\text{Cnv}ʻʻ\omega\), and is non-inductive and reflexive. (4) The series may be open and have neither beginning nor end. This occurs if \[ {\sim}(xP_{\text{po}}x).{\sim}\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.{\sim}\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx \] In this case we get a series whose relation-number is the sum (in the sense of *180) of \(\text{Cnv}ʻʻ\omega\) and \(\omega\), which again is non-inductive and reflexive. In all four cases, if \(y\) and \(z\) be any two members of the family of \(x\), the interval between \(y\) and \(z\) is an inductive class.

If \(x\) is a member of \(\overrightarrow{B}ʻP\), or if the family of \(x\) contains a member of \(\overrightarrow{B}ʻP\), cases (1) and (4) are excluded, since the series has a beginning. In this case the number of predecessors of any term is an inductive number. It will be observed that every family is either wholly contained in \(sʻ\text{gen}ʻP\) or wholly contained in \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\); families of kinds (2) and (3) (excluding, in (2), those which have an end but no beginning) are contained in \(sʻ\text{gen}ʻP\), while families of kinds (1) and (4), and those of (2) which have an end but no beginning, are contained in \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\); families containing a member of \(\overrightarrow{B}ʻP\) are contained in \(sʻ\text{gen}ʻP\), while all others are contained in \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\).

Thus a one-one relation in general gives rise to a number of wholly disconnected series, some closed, others open and with or without a beginning or an end. The condition that all the series should be open is \(P_{\text{po}}\,\unicode{x2abd}\, J\).

The case of a \(Q\)-shaped family, considered in *96, cannot arise when \(P\in 1\rightarrow 1\), for in a \(Q\)-shaped family the term at the junction of the tail and the circle has two predecessors, one in the tail and one in the circle, so that the relation in question is not \(1\rightarrow 1\). It follows that, when \(P\in 1\rightarrow 1\), if \(\alpha\) is a family containing a member of \(\overrightarrow{B}ʻP\), \(\alpha \upharpoonleft P_{\text{po}}\,\unicode{x2abd}\, J\) (cf. *96·23).

When \(BʻP\) exists, there is only one family which has a beginning. In this case, ignoring the other families (if any), we call the members of the family of \(BʻP\) respectively \(1_{P}\), \(2_{P}\), \(3_{P}\), .... If the family has \(\nu\) members, where \(\nu\) is an inductive cardinal, its last member will be \(\nu_{P}\). If on the other hand the number of members of the family is not an inductive cardinal, it must be \(\aleph _{0}\); in this case, the family forms a progression, whose members are \(1_{P}\), \(2_{P}\), \(3_{P}\), ..., \(\nu _{P}\), ..., where \(\nu _{P}\) always exists when \(\nu\) is an inductive cardinal.

[Pg 256]

In addition to the propositions already mentioned, the following are important:

*122·21. \(\vdash \colon\ldotp R\in \text{Prog}.x,y\in CʻR.\supset :xR_{\text{po}}y.\lor.x = y.\lor.yR_{\text{po}}x\)

(Cf. note to *122·21, below.)

*122·34. \(\vdash \colon\ldotp R\in \text{Prog}.\supset :\nu \in \text{NC induct} - \iota ʻ0.\equiv .\text{E}!\nu _{R}\)

*122·341. \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR = \hat{x} \{(\exists \nu ).\nu \in \text{NC induct} - \iota ʻ0.x = \nu _{R}\}\)

In virtue of these two propositions, the terms of a progression are \[ 1_{R},\,\, 2_{R},\,\, 3_{R},\,\, ... \nu _{R},\,\, ..., \] where every inductive cardinal occurs. This is the same fact as is usually assumed when the terms are represented as \[ x_{1},\,\, x_{2},\,\, x_{3},\,\, ... x_{\nu }, .... \]

*122·35. \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct} - \iota ʻ0.\supset .\overrightarrow{B}ʻR_{\nu } = R(1_{R}\vdash\dashv \nu _{R}).\overrightarrow{B}ʻR_{\nu }\in \nu \)

*122·36. \(\vdash :\exists !\text{Prog} \cap t^{11}ʻx.\supset .\text{Infin ax}(x)\)

*122·37. \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR{\sim}\in \text{Cls induct}.\text{N}_{0}\text{c}ʻ\text{D}ʻR{\sim}\in \text{NC induct}\)

*122·38. \(\vdash :R\in \text{Prog}.\supset .\overrightarrow{R}_{*}ʻx\in \text{Cls induct}\)

I.e. the number of terms up to any given point of a progression is inductive.

*122·01. \(\text{Prog} = (1\rightarrow 1)\cap \hat{R} (\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR) \quad\text{Df}\)

*122·1. \(\vdash :R\in \text{Prog}.\equiv .R\in 1\rightarrow 1.\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR \quad[(*122·01)]\)

*122·11. \(\vdash \colon\ldotp R\in \text{Prog}.\equiv :R\in 1\rightarrow 1.\text{E}!BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻBʻR\)

Dem. \[ \begin{array}{l} \vdash .*122·1.*14·205.\supset \\ \vdash \colon\colon R\in \text{Prog}.&\equiv \colon\ldotp R\in 1\rightarrow 1:(\exists a).a = BʻR.\text{D}ʻR = \overleftarrow{R}_{*}ʻa\colon\ldotp \\ [*20·43] &\equiv \colon\ldotp R\in 1\rightarrow 1\colon\ldotp (\exists a):a = BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻa\colon\ldotp \\ [*14·15] &\equiv \colon\ldotp R\in 1\rightarrow 1\colon\ldotp (\exists a):a = BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻBʻR\colon\ldotp \\ [*14·204] &\equiv \colon\ldotp R\in 1\rightarrow 1.\text{E}!BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻBʻR\colon\colon \supset \vdash .\text{Prop} \end{array} \]

Observe that, by the conventions as to descriptive symbols, \(\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR\) involves the existence of \(BʻR\), whereas \(x\in \text{D}ʻR.\equiv _{x}.x\in\overleftarrow{R}_{*}ʻBʻR\) does not, since, if \(BʻR\) does not exist, we have \((x).x{\sim}\in \overleftarrow{R}_{*}ʻBʻR\), and therefore \((x).x{\sim}\in \text{D}ʻR\) will satisfy the equivalence, i.e. \(\dot{\Lambda}\) will satisfy the equivalence although it has no first term. This is the reason why \(\text{E}!BʻR\) appears explicitly in *122·11, though it was only implicit in *122·1.

*122·12. \[\begin{align}\vdash \colon\colon R\in \text{Prog}.&\equiv \colon\ldotp R\in 1\rightarrow 1.\text{E}!BʻR\colon\ldotp x\in \text{D}ʻR.\equiv _{x}:\\ &BʻR\in \alpha .\breve{R} ʻʻ\alpha \subset \alpha .\supset _{\alpha }.x\in \alpha \quad[*122·11.*90·1]\end{align}\]

[Pg 257]

*122·14. \(\vdash :R\in \text{Prog}.\supset .\overleftarrow{R}_{\text{po}}ʻBʻR=\text{ᗡ}ʻR\)

Dem. \[ \begin{array}{l} \vdash .*122·1.*37·25. \supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻR&=\breve{R} ʻʻ\overleftarrow{R}_{*}ʻBʻR\\ [*91·52] & =\overleftarrow{R}_{\text{po}}ʻBʻR:\supset \vdash .\text{Prop} \end{array} \]

*122·141. \(\vdash :R\in \text{Prog}.\supset .\text{ᗡ}ʻR\subset \text{D}ʻR.CʻR=\text{D}ʻR\)

Dem. \[ \begin{array}{l} \vdash .*122·1.*37·25. \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻR=\breve{R} ʻʻ\overleftarrow{R}_{*}ʻBʻR.\\ [*90·163] &\supset .\text{ᗡ}ʻR\subset \overleftarrow{R}_{*}ʻBʻR.\\ [*122·1.*33·181] &\supset .\text{ᗡ}ʻR\subset \text{D}ʻR.CʻR=\text{D}ʻR:\supset \vdash .\text{Prop} \end{array} \]

*122·142. \(\vdash :R\in \text{Prog}.P\in \text{Pot}ʻR.\supset .\text{D}ʻP=\text{D}ʻR \quad[*122·141.*92·14]\)

*122·143. \(\vdash :R\in \text{Prog}.P\in \text{Pot}ʻR.\supset .\text{ᗡ}ʻP\subset \text{D}ʻP \quad[*122·142·141.*91·271]\)

*122·15. \(\vdash :R\in \text{Prog}.\supset .R=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R=R\upharpoonright (\overleftarrow{R}_{\text{po}}ʻBʻR)=R\upharpoonright (\overleftarrow{R}_{*}ʻBʻR)\)

Dem. \[ \begin{array}{l} \vdash .*122·1.*35·63. \supset \vdash :\text{Hp}.\supset .R&=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R\\ [*96·2] &=R\upharpoonright (\overleftarrow{R}_{\text{po}}ʻBʻR)\\ [*96·21] &=R\upharpoonright (\overleftarrow{R}_{*}ʻBʻR):\supset \vdash .\text{Prop} \end{array} \]

*122·151. \[\begin{align}&\vdash :R\in \text{Prog}.\supset .R_{*}=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R_{*}=R_{*}\upharpoonright (\overleftarrow{R}_{*}ʻBʻR)\\ &[35·63·66.*90·14.*122·141·1]\end{align}\]

*122·152. \[\begin{align}\vdash :R\in \text{Prog}.\supset .R_{\text{po}}=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R_{\text{po}}&=R_{\text{po}}\upharpoonright (\overleftarrow{R}_{\text{po}}ʻBʻR)\\ &=R_{\text{po}}\upharpoonright (\overleftarrow{R}_{*}ʻBʻR)\\ [*35·63·66.*91·504.*121·1·14]\end{align}\]

*122·16. \(\vdash :R\in \text{Prog}.\supset .R_{\text{po}}\,\unicode{x2abd}\, J \quad[*96·23.*122·152]\)

This proposition enables us to apply to progressions all the propositions of *121 in which we have as hypothesis \[ R\in \text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J, or R\in 1\rightarrow \text{Cls}.R_{\text{po}}\,\unicode{x2abd}\, J. \]

*122·17. \(\vdash :R\in \text{Prog}.\equiv .R\in \text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J.\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR\)

Dem. \[ \begin{array}{l} \vdash .*35·63. \supset \vdash :\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR.\supset .R=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R &\qquad \text{(1)}\\ \vdash .*96·453. \supset \vdash :R\in \text{Cls}\rightarrow 1.(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R_{\text{po}}\,\unicode{x2abd}\, J.\supset .(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R\in 1\rightarrow 1 &\qquad \text{(2)}\\ \vdash .(1).(2).*122·1. \supset \vdash :\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR.R\in \text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J.\supset .R\in \text{Prog} &\qquad \text{(3)}\\ \vdash .(3).*122·1·16.\supset \vdash .\text{Prop} \end{array} \]

[Pg 258]

To illustrate this proposition, consider its application to the inductive cardinals arranged in order of magnitude; i.e. take as a value of \(R\) the relation \[ \hat{\mu} \hat{\nu} (\mu \in \text{NC induct}.\nu =\mu +_{c}1). \]

We then have \(R\in \text{Cls}\rightarrow 1.0=BʻR\); also \[ \text{NC induct}=\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR. \]

We have also \[ \exists !\mu +_{c}1.\mu +_{c}1=\nu +_{c}1.\supset .\mu =\nu , \] so that \(R\upharpoonright (-\iota ʻ\Lambda )\in 1\rightarrow \text{Cls}\).

Again \[ \mu R_{\text{po}}\nu .\equiv .(\exists \varpi ).\varpi \in \text{NC induct}-\iota ʻ0.\nu =\mu +_{c}\varpi , \] whence \[ \mu R_{\text{po}}\nu .\exists !\mu .\supset .\mu \neq \nu , \] i.e. \[ (-\iota ʻ\Lambda )\upharpoonleft R_{\text{po}}\,\unicode{x2abd}\, J. \]

But we do not get \(R\in 1\rightarrow \text{Cls}\) or \(R_{\text{po}}\,\unicode{x2abd}\, J\) unless we have \[ \Lambda {\sim}\in \text{NC induct}, \] which is the axiom of infinity. If this condition fails, we reach at last an inductive cardinal which = \(\Lambda\), and we have \[ \Lambda =\Lambda +_{c}1, \] so that \(\Lambda\) has two immediate predecessors, namely itself and the last existent cardinal. The posterity of 0, in this case, is a \(Q\) in which the circle has narrowed to a single term, namely \(\Lambda\).

Thus we need the axiom of infinity in order to prove \[ \hat{\mu} \hat{\nu} (\mu \in \text{NC induct}.\nu =\mu +_{c}1)\in \text{Prog}. \]

*122·2. \(\vdash \colon\ldotp R\in \text{Prog}.x,y\in CʻR.\supset :xR_{*}y.\lor.yR_{*}x \quad[*96·302.*122·1·141]\)

*122·21. \[\begin{align}\vdash \colon\ldotp R\in \text{Prog}.x,y\in CʻR.\supset :xR_{\text{po}}y.&\lor.x=y.\lor.yR_{\text{po}}x\\ &[*96·303.*122·1·141]\end{align}\]

This proposition, together with *122·16 and *91·56, shows that if \(R\in \text{Prog}\), \(R_{\text{po}}\) has the three properties by which transitive serial relations are defined (cf. *204), namely it is (1) transitive, (2) contained in diversity, (3) connected, i.e. such that it relates any two distinct members of its field. We shall at a later stage define the ordinal number \(\omega\) as the class of such relations as \(R_{\text{po}}\), where \(R\in \text{Prog}\).

*122·22. \(\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.x,y\in \alpha -\breve{R} _{\text{po}}ʻʻ\alpha .\supset .x=y\)

Dem. \[ \begin{array}{l} \vdash .*122·21.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xR_{\text{po}}y.\lor.x=y.\lor.yR_{\text{po}}x &\qquad \text{(1)}\\ \vdash .*37·105.&\supset \vdash :x\in \alpha .xR_{\text{po}}y.\supset .y\in \breve{R} _{\text{po}}ʻʻ\alpha :\\ [\text{Transp}] &\supset \vdash :x\in \alpha .y{\sim}\in \breve{R} _{\text{po}}ʻʻ\alpha .\supset .{\sim}(xR_{\text{po}}y) &\qquad \text{(2)}\\ \vdash .(2). &\supset \vdash :\text{Hp}.\supset .{\sim}(xR_{\text{po}}y).{\sim}(yR_{\text{po}}x) &\qquad \text{(3)}\\ \vdash .(1).(3).&\supset \vdash .\text{Prop} \end{array} \]

[Pg 259]

*122·23. \[\begin{align}\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.&\exists !\alpha .\supset .\\ &\text{E}!\text{min}(R_{\text{po}})ʻ\alpha .\alpha -\breve{R} _{\text{po}}ʻʻ\alpha =\iota ʻ\text{min}(R_{\text{po}})ʻ\alpha\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*96·52. &\supset \vdash :\text{Hp}.\supset .\exists !\overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha &\qquad \text{(1)}\\ \vdash .*93·111.*122·22. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x,y\in \overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha .\supset _{x,y}.x=y &\qquad \text{(2)}\\ \vdash .(1).(2).*32·4.*93·111.\supset \vdash .\text{Prop} \end{array} \]

This proposition shows that every existent class contained in a progression has a first term, i.e. that a progression is a well-ordered series (cf. *250).

*122·231. \(\vdash :R\in \text{Prog}.\alpha \subset \breve{R} _{\text{po}}ʻʻ\alpha .\supset .\alpha =\Lambda\)

Dem. \[ \begin{array}{l} \vdash .*91·504.&\supset \vdash :\text{Hp}.\supset .\alpha \subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ \vdash .*93·11. &\supset \vdash :\text{Hp}.\supset.{\sim}\text{E}!\overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*122·23·141.\text{Transp}.\supset \vdash .\text{Prop} \end{array} \]

*122·24. \(\vdash :R\in \text{Prog}.P\in PotʻR.\supset .\text{D}ʻP=\breve{P} _{*}ʻʻ\overrightarrow{B}ʻP=sʻ\text{gen}ʻP\)

Dem. \[ \begin{array}{l} \vdash .*122·1.*92·102.\supset \vdash :\text{Hp}.&\supset .P\in 1\rightarrow 1.\\ [*93·42] &\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=\breve{P} ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.\\ [*91·581] &\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\subset \breve{R} _{\text{po}}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.\\ [*122·231] &\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=\Lambda &\qquad \text{(1)}\\ \vdash .(1).*93·37·36. \supset \vdash :\text{Hp}.&\supset .CʻP=\breve{P} _{*}ʻʻ\overrightarrow{B}ʻP=sʻ\text{gen}ʻP &\qquad \text{(2)}\\ \vdash .(2).*122·143.\supset \vdash .\text{Prop} \end{array} \]

Except when \(P=R\), \(\overrightarrow{B}ʻP\) will not reduce to a single term. In fact, if \(P=R_{\nu }\), \(\overrightarrow{B}ʻP=R(1_{R}\vdash\dashv \nu _{R})\), i.e. \(\overrightarrow{B}ʻP\) consists of the first \(\nu\) terms of the progression.

*122·25.\[\begin{align}\vdash :R\in \text{Prog}.P\in \text{Pot}ʻR.x\in &\text{D}ʻR.\supset .\\ &(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\in \text{Prog}.x=Bʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*122·1.*92·102.&\supset \vdash :\text{Hp}.\supset .(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\in 1\rightarrow 1 &\qquad \text{(1)}\\ \vdash .*122·143.& \supset \vdash :\text{Hp}.\supset .\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP.\\ [*35·62] &\supset .\text{D}ʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\}=\overleftarrow{P}_{*}ʻx &\qquad \text{(2)}\\ \vdash .*37·4.*91·52. &\supset \vdash .\text{ᗡ}ʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\}=\overleftarrow{P}_{\text{po}}ʻx &\qquad \text{(3)}\\ \vdash .*122·16.*91·6. &\supset \vdash :\text{Hp}.\supset .x{\sim}\in \overleftarrow{P}_{\text{po}}ʻx &\qquad \text{(4)}\\ \vdash .*91·542. &\supset \vdash :y\in \overleftarrow{P}_{*}ʻx.y\neq x.\supset .y\in \overleftarrow{P}_{\text{po}}ʻx &\qquad \text{(5)}\\ \vdash .(2).(3).(4).(5).&\supset \vdash :\text{Hp}.\supset .x=Bʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\} &\qquad \text{(6)}\\ \vdash .(1).(2).(6).*96·131.\supset \vdash .\text{Prop} \end{array} \]

[Pg 260]

The above proposition shows that what we may call an "arithmetical progression" in a progression is a progression, i.e. if, starting from any term of a progression, we take every other term, or every third term, or every \(\nu\)th term, we still have a progression.

*122·26. \(\vdash :R\in \text{Prog}.\alpha \subset R_{\text{po}}ʻʻ\alpha .\exists !\alpha .\supset .\text{D}ʻR=R_{\text{po}}ʻʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash . *22·1 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : BʻR \in \alpha . \supset BʻR \in R_{\text{po}}ʻʻ\alpha &\qquad \text{(1)}\\ \vdash .*91·542.*122·11.&\supset \vdash \colon\ldotp \text{Hp}.BʻR{\sim}\in \alpha .\supset :y\in \alpha \cap \text{D}ʻR.\supset _{y}.(BʻR)R_{\text{po}}y:\\ [*91·504.*37·15] &\supset :y\in \alpha .\supset _{y}.(BʻR)R_{\text{po}}y:\\ [*10·55.\text{Hp}] &\supset :(\exists y).y\in \alpha .(BʻR)R_{\text{po}}y:\\ [*37·1] &\supset :BʻR\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2) .&\supset \vdash :\text{Hp}.\supset .BʻR\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(3)}\\ \vdash .*92·111. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in R_{\text{po}}ʻʻ\alpha .xRy.\supset .y\in R_{*}ʻʻ\alpha .\\ [*91·545] &\supset .y\in \alpha \cup R_{\text{po}}ʻʻ\alpha .\\ [\text{Hp}] &\supset .y\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(4)}\\ \vdash .(3).(4).*90·112.&\supset \vdash \colon\ldotp \text{Hp}.\supset :(BʻR)R_{*}y.\supset .y\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(5)}\\ \vdash .(5).*122·1.\supset \vdash .\text{Prop} \end{array} \]

The above proposition shows that if an existent class contained in a progression has no maximum, then any assigned member of the progression is succeeded by members of the class.

The following proposition states that if \(\alpha\) has members belonging to a progression, and there are members of the progression which do not precede any member of \(\alpha\), then there is in the progression a last member of \(\alpha\).

*122·27. \[\begin{align}\vdash :R\in \text{Prog}.\exists !\text{D}ʻR-R_{\text{po}}ʻʻ\alpha .&\exists !\alpha \cap \text{D}ʻR.\supset .\\ &\text{E}!\text{max}(R_{\text{po}})ʻ\alpha .\exists !\alpha \cap \text{D}ʻR-R_{\text{po}}ʻʻ\alpha \end{align}\]

Dem. \[ \begin{array}{l} \vdash .*122·26.\text{Transp}.*37·265. &\supset \vdash :\text{Hp}.\supset .\exists !\alpha \cap CʻR-R_{\text{po}}ʻʻ\alpha &\qquad \text{(1)}\\ \vdash .*122·21. &\supset \vdash :\text{Hp}.x,y\in \alpha \cap CʻR-R_{\text{po}}ʻʻ\alpha .\supset .x=y &\qquad \text{(2)}\\ \vdash .(1).(2).*93·115.*122·141. &\supset \vdash .\text{Prop} \end{array} \]

*122·28. \(\vdash :R\in \text{Prog}.\alpha \subset \overrightarrow{R}_{*}ʻx.\exists !\alpha .\supset .\text{E}!\text{max}(R_{\text{po}})ʻ\alpha .\exists !\alpha \cap \text{D}ʻR-R_{\text{po}}ʻʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*90·13.*122·141. &\supset \vdash :\text{Hp}.\supset .\alpha \subset \text{D}ʻR &\qquad \text{(1)}\\ \vdash .*90·14.*122·141. &\supset \vdash :\text{Hp}.\supset .x\in \text{D}ʻR.\\ [*71·161.*122·16] &\supset .\breve{R} ʻx{\sim}\in R_{\text{po}}ʻʻ\alpha .\\ [*122·1] &\supset .\exists !\text{D}ʻR-R_{\text{po}}ʻʻ\alpha &\qquad \text{(2)}\\ \vdash .(1).(2).*122·27.\supset \vdash .\text{Prop} \end{array} \]

[Pg 261]

*122·3. \[\begin{align}&\vdash :R\in \text{Prog}.\supset .\text{D}ʻR=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}.(BʻR)R_{v}x\}\\ &[*121·52.*122·1·16]\end{align}\]

*122·31. \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\text{ᗡ}ʻR_{\nu }=\hat{y} \{\text{Nc}ʻR(BʻR\vdash\dashv y)>\nu\}\)

Dem. \[ \begin{array}{l} \vdash .*120·429.\supset \vdash \colon\ldotp \text{Hp}.\supset :&\text{Nc}ʻR(BʻR\vdash\dashv y)>\nu .\equiv .\text{Nc}ʻR(BʻR\vdash\dashv y)\geq \nu +_{c}1.\\ [*117·31] &\equiv .(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻR(BʻR\vdash\dashv y)=\mu +_{c}\nu +_{c}1.\\ &\nu +_{c}1,\mu +_{c}\nu +_{c}1\in \text{N}_{0}\text{C}.\\ [*121·45.*120·452.*110·4] &\equiv .(\exists \mu ).\mu \in \text{NC induct}.\\ &\text{Nc}ʻR(BʻR\vdash\dashv y)=\mu +_{c}\nu +_{c}1.\mu +_{c}\nu +_{c}1\in \text{N}_{0}\text{C}.\\ [*121·11·35.*110·43.*100·3] &\equiv .(\exists \mu ).\mu \in \text{NC induct}.(BʻR)R_{\mu }\mid R_{\nu }y.\\ [*34·1] &\equiv .(\exists \mu ,x).\mu \in \text{NC induct}.(BʻR)R_{\mu }x.xR_{\nu }y.\\ [*122·3] &\equiv .(\exists x).x\in \text{D}ʻR.xR_{\nu }y.\\ [*121·323] &\equiv .(\exists x).xR_{\nu }y.\\ [*33·131] &\equiv .y\in \text{ᗡ}ʻR_{\nu }\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*122·32. \[\begin{align}\vdash :R\in \text{Prog}.\nu \in &\text{NC induct}-\iota ʻ0.\supset .\\ &\overrightarrow{B}ʻR_{\nu }=\text{D}ʻR\cap \hat{x} \{\text{Nc}ʻR(BʻR\vdash\dashv x)\leq \nu\}\end{align}\]

Dem. \[ \begin{array}{l} \vdash .*122·142.*121·501.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻR_{\nu }=\text{D}ʻR &\qquad \text{(1)}\\ \vdash .*122·31.*120·442.&\supset \vdash :\text{Hp}.\supset .-\text{ᗡ}ʻR_{\nu }=\hat{x} \{\text{Nc}ʻR(BʻR\vdash\dashv x)\leq \nu\} &\qquad \text{(2)}\\ \vdash .(1).(2).*93·101.\supset \vdash .\text{Prop} \end{array} \]

*122·33. \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\text{E}!\nu _{R}\)

Dem. \[ \begin{array}{l} \vdash .*121·601.*122·11. &\supset \vdash :\text{Hp}.\supset .\text{E}!1_{R} &\qquad \text{(1)}\\ \vdash .*121·634·637.*122·141.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\nu _{R}.\supset .\text{E}!(\nu +_{c}1)_{R} &\qquad \text{(2)}\\ \vdash .(1).(2).*120·473.\supset \vdash .\text{Prop} \end{array} \]

*122·34. \(\vdash \colon\ldotp R\in \text{Prog}.\supset :\nu \in \text{NC induct}-\iota ʻ0.\equiv .\text{E}!\nu _{R} \quad[*122·33.*121·635]\)

*122·341. \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ0.x=\nu _{R}\}\)

Dem. \[ \begin{array}{l} \vdash .*122·3·34.*121·638.\supset \\ \vdash :\text{Hp}.\supset .\text{D}ʻR&=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}.x=(\nu +_{c}1)_{R}\}\\ [*120·471] &=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ0.x=\nu _{R}\}:\supset \vdash .\text{Prop} \end{array} \]

[Pg 262]

In virtue of *122·34·341, all the terms of a progression occur in the series \(1_{R}\), \(2_{R}\), ... \(\nu _{R}\), ..., and every inductive cardinal except 0 is used in forming this series.

*122·35. \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\overrightarrow{B}ʻR_{\nu }=R(1_{R}\vdash\dashv \nu _{R}).\overrightarrow{B}ʻR_{\nu }\in \nu\)

Dem. \[ \begin{array}{l} \vdash .*121·63.*122·33.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻR(BʻR\vdash\dashv \nu _{R})=\nu . &\qquad \text{(1)}\\ [*122·32] \supset .\overrightarrow{B}ʻR_{\nu }&=\text{D}ʻR\cap \hat{x} \{\text{Nc}ʻR(BʻR\vdash\dashv x)\leq \text{Nc}ʻR(BʻR\vdash\dashv \nu _{R})\}\\ [*121·481] &=\text{D}ʻR\cap \hat{x} \{R(BʻR\vdash\dashv x)\subset R(BʻR\vdash\dashv \nu _{R})\}\\ [*122·1.*121·103] &=\hat{x} \{(BʻR)R_{*}x:yR_{*}x.\supset _{y}.yR_{*}\nu _{R}\}\\ [*90·17·13.*10·1] &=\hat{x} \{(BʻR)R_{*}x.xR_{*}\nu _{R}\}\\ [*121·103] &=R(BʻR\vdash\dashv \nu _{R}) &\qquad \text{(2)}\\ [*121·601.*122·11] &=R(1_{R}\vdash\dashv \nu _{R}) &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*122·36. \(\vdash :\exists !\text{Prog}\cap t^{11}ʻx.\supset .\text{Infin ax}(x)\)

Dem. \[ \begin{array}{l} \vdash .*122·35. &\supset \vdash :R\in \text{Prog}\cap t^{11}ʻx.\nu \in \text{NC induct}-\iota ʻ0.\supset .\exists !\nu (x) &\qquad \text{(1)}\\ \vdash .(1).*101·12. &\supset \vdash \colon\ldotp R\in \text{Prog}\cap t^{11}ʻx.\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\nu (x):\\ [*120·301] &\supset :\text{Infin ax}(x)\colon\ldotp \supset \vdash .\text{Prop} \end{array} \]

*122·37. \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR{\sim}\in \text{Cls induct}.\text{N}_{0}\text{c}ʻ\text{D}ʻR{\sim}\in \text{NC induct}\)

Dem. \[ \begin{array}{l} \vdash .*122·35. \supset \vdash \colon\ldotp R\in \text{Prog}.&\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\text{Cl}ʻ\text{D}ʻR\cap (\nu +_{c}1).\\ [*117·22·107] &\supset _{\nu }.\text{N}_{0}\text{c}ʻ\text{D}ʻR\geq \nu +_{c}1.\\ [*120·429] &\supset _{\nu }.\text{N}_{0}\text{c}ʻ\text{D}ʻR>\nu .\\ [*117·42] &\supset _{\nu }.\text{N}_{0}\text{c}ʻ\text{D}ʻR\neq \nu :\\ [*13·196] &\supset :\text{N}_{0}\text{c}ʻ\text{D}ʻR{\sim}\in \text{NC induct} &\qquad \text{(1)}\\ \vdash .(1).*120·21.\supset \vdash .\text{Prop} \end{array} \]

*122·38. \(\vdash :R\in \text{Prog}.\supset .\overrightarrow{R}_{*}ʻx\in \text{Cls induct} \quad[*121·7.*90·13.*120·212]\)

*122·381. \[\begin{align}&\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\overrightarrow{R}_{*}ʻ\nu _{R}=R(1_{R}\vdash\dashv \nu _{R}).\overrightarrow{R}_{*}ʻ\nu _{R}\in \nu \\ &[*121·7.*122·35]\end{align}\]

[Pg 263]

The following series of propositions are concerned in proving that any class contained in a progression is inductive if it has a last term, and is a progression if it has no last term. In the latter case, it is supposed arranged in the same order as it had in the original progression. A certain complication is necessary in order to define its one-one generating relation. If \(R\) is the generating relation of the original progression, we proceed first to \(R_{\text{po}}\), then to \(R_{\text{po}}\unicode{x0294f}\alpha\), where \(\alpha\) is the class in question; this gives us a transitive generating relation for \(\alpha\). Calling this relation \(P\), we then proceed to \(P\dot{-} P^{2}\), i.e. the relation of consecutive members of the series generated by \(P\). This relation turns out to be one-one, and to arrange \(\alpha\) in a progression; hence our proposition is proved. The reason for the necessity of this detour is that consecutive members of \(\alpha\) may not be consecutive members of the original progression.

*122*41. \(\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.y\in \alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \subset R(BʻR\vdash\dashv y)\)

Dem. \[ \begin{array}{l} \vdash .*37·1.*10·51.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \alpha .\supset _{z}.{\sim}(yR_{\text{po}}z).\\ [*122·21] &\supset _{z}.zR_{*}y &\qquad \text{(1)}\\ \vdash .*122·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \alpha .\supset _{z}.(BʻR)R_{*}z &\qquad \text{(2)}\\ \vdash .(1).(2).*121·103.\supset \vdash .\text{Prop} \end{array} \]

*122·42. \(\vdash :R\in \text{Prog}.\alpha \subset R(BʻR\vdash\dashv y).y\in \alpha .\supset .y= \text{max}_{R}ʻ\alpha\)

Dem. \[ \begin{array}{l} \vdash .*121·103.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \alpha .\supset _{z}.zR_{*}y.\\ [*91·574.*122·16] &\supset _{z}.{\sim}(yR_{\text{po}}z):\\ [*37·1.*10·51] &\supset :y{\sim}\in R_{\text{po}}ʻʻ\alpha : &\qquad \text{(1)}\\ [*96·303] &\supset :z\in \alpha -R_{\text{po}}ʻʻ\alpha .\supset _{z}.z=y &\qquad \text{(2)}\\ \vdash .(1).(2).*93·115.\supset \vdash .\text{Prop} \end{array} \]

*122·43. \[\begin{align}&\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.\exists !\alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \in \text{Cls induct}\\ &[*122·41.*121·45.*120·481]\end{align}\]

Thus every class which is contained in a progression and has a last term is inductive. We have next to prove \[ R\in \text{Prog} .\alpha \subset \text{D}ʻR.\exists !\alpha .{\sim}\exists !\alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \in \text{D}ʻʻ\text{Prog}. \] This is effected in the following propositions.

*122·44. \[\begin{align}\vdash :R\in \text{Prog} .\alpha \subset R_{\text{po}}ʻʻ\alpha .\exists !\alpha .P=R_{\text{po}}\unicode{x0294f}\alpha .Q&=P\dot{-} P^{2}.\supset .\\ &Q\in 1\rightarrow 1.Q\,\unicode{x2abd}\, R_{\text{po}}\end{align}\]

Note. The hypothesis here exceeds what is necessary for the conclusion, but is the hypothesis required for *122·45, for which the present and the following propositions are lemmas.

Dem. \[ \begin{array}{l} \vdash .*23·43.*35·442.\supset \vdash :\text{Hp}.&\supset .Q\,\unicode{x2abd}\, R_{\text{po}} &\qquad \text{(1)}\\ \vdash .*36·13. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x,y,z\in \alpha .xR_{\text{po}}y.yR_{\text{po}}z.\supset .xP^{2}z:\\ [\text{Transp}] &\supset :x,y,z\in \alpha .xR_{\text{po}}y.{\sim}(xP^{2}z).\supset .{\sim}(yR_{\text{po}}z):\\ [*36·13] &\supset :xPy.{\sim}(xP^{2}z).\supset .{\sim}(yR_{\text{po}}z):\\ [*3·47] &\supset :xQy.xQz.\supset .{\sim}(yR_{\text{po}}z).{\sim}(zR_{\text{po}}y).\\ [*122·21.(1)] &\supset .y=z &\qquad \text{(2)}\\ \text{Similarly}\quad \vdash \colon\ldotp \text{Hp}.&\supset :xQz.yQz.\supset .x=y &\qquad \text{(3)}\\ \vdash .(1).(2).(3).\supset \vdash .\text{Prop} \end{array} \]

*122·441. \(\vdash :\text{Hp}*122·44.\supset .\text{D}ʻQ=\alpha\)

Dem. \[ \begin{array}{l} \vdash .*37·41.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻQ\subset \alpha &\qquad \text{(1)}\\ \vdash .*37·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \alpha .\supset .(\exists y).y\in \alpha .xR_{\text{po}}y.\\ [*36·13] &\supset .\exists !\overleftarrow{P}ʻx.\\ [*122·23.*93·11] &\supset .\exists !\overleftarrow{P}ʻx-\breve{R} _{\text{po}}ʻʻ\overleftarrow{P}ʻx.\\ [*35.442] &\supset .\exists !\overleftarrow{P}ʻx-\breve{P} ʻʻ\overleftarrow{P}ʻx.\\ [*37·311.*32·31·35] &\supset .\exists !\overleftarrow{Q}ʻx &\qquad \text{(2)}\\ \vdash .(1).(2).*33·4.\supset \vdash .\text{Prop} \end{array} \]

[Pg 264]

*122*442. \(\vdash :\text{Hp}*122·44.\supset .P=Q_{\text{po}}\)

In proving \(P\,\unicode{x2abd}\, Q_{\text{po}}\) below, we assume \(xPz\) and consider the maximum of \(\overrightarrow{R}_{\text{po}}ʻz\cap\overleftarrow{Q}_{*}ʻx\), which is shown to exist and be \(Qʻz\), whence \(xQ_{\text{po}}z\).

Dem. \[ \begin{array}{l} \vdash . *23·43.&\supset \vdash :\text{Hp}.\supset .Q\,\unicode{x2abd}\, P &\qquad \text{(1)}\\ \vdash . *91·56.&\supset \vdash \colon\ldotp \text{Hp}.\supset :P^{2}\,\unicode{x2abd}\, P:\\ [(1)] &\supset :S\,\unicode{x2abd}\, P.\supset .S\mid Q\,\unicode{x2abd}\, P &\qquad \text{(2)}\\ \vdash .(1).(2).*91·171.*41·151.&\supset \vdash \colon\ldotp \text{Hp}.\supset .Q_{\text{po}}\,\unicode{x2abd}\, P &\qquad \text{(3)}\\ \vdash .*36·13.*121·1.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xP^{2}z.\equiv .x,z\in \alpha .\exists !\alpha \cap R(x-z):\\ [\text{Transp}.\text{Fact}] &\supset :xQz.\equiv .x,z\in \alpha .xR_{\text{po}}z.\alpha \cap R(x-z)=\Lambda &\qquad \text{(4)}\\ \vdash .*122·441.&\supset \vdash \colon\ldotp \text{Hp}.xPz.\supset :x\in (\overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx):\\ [*122·27] &\supset :\exists !\overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx-R_{\text{po}}ʻʻ(\overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx):\\ [*37·461] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx=\Lambda :\\ [*90·151] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}ʻy=\Lambda :\\ [(4)] &\supset :(\exists y):y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx:\\ &{\sim}(\exists w).w\in \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz.\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻw=\Lambda :\\ [*22·43.*91·56] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx:\\ &{\sim}(\exists w).w\in \alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz.\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz\cap \overrightarrow{R}_{\text{po}}ʻw=\Lambda :\\ [*37·461] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\\ &{\sim}\exists !\alpha \cap :\overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz-\breve{R} _{\text{po}}ʻʻ(\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz):\\ [*122*28.\text{Transp}] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz=\Lambda :\\ [(4)] &\supset :(\exists y).y\in \overleftarrow{Q}_{*}ʻx.yQz:\\ [*91·52] &\supset :xQ_{\text{po}}z &\qquad \text{(5)}\\ \vdash .(3).(5).\supset \vdash .\text{Prop} \end{array} \]