Principia Mathematica Vol. 2

PRINCIPIA MATHEMATICA

CONTENTS OF VOLUME II

	PAGE
PREFATORY STATEMENT OF SYMBOLIC CONVENTIONS	ix
PART III. CARDINAL ARITHMETIC.
Summary of Part III	3
SECTION A. DEFINITION AND LOGICAL PROPERTIES OF CARDINAL NUMBERS	4
*100. Definition and elementary properties of cardinal numbers	13
*101. On 0 and 1 and 2	19
*102. On cardinal numbers of assigned types	24
*103. Homogeneous cardinals	36
*104. Ascending cardinals	42
*105. Descending cardinals	52
*106. Cardinals of relational types	60
SECTION B. ADDITION, MULTIPLICATION AND EXPONENTIATION	66
*110. The arithmetical sum of two classes and of two cardinals	75
*111. Double similarity	88
*112. The arithmetical sum of a class of classes	97
*113. On the arithmetical product of two classes or of two cardinals	105
*114. The arithmetical product of a class of classes	124
*115. Multiplicative classes and arithmetical classes	135
*116. Exponentiation	143
*117. Greater and less	171
General note on cardinal correlators	185
SECTION C. FINITE AND INFINITE	187
*118. Arithmetical substitution and uniform formal numbers	193
*119. Subtraction	201
*120. Inductive cardinals	207
*121. Intervals	233
*122. Progressions	253
*123.	268
*124. Reflexive classes and cardinals	278
*125. The axiom of infinity	289
*126. On typically indefinite inductive cardinals	293
PART IV. RELATION-ARITHMETIC.
Summary of Part IV	301
SECTION A. ORDINAL SIMILARITY AND RELATION-NUMBERS	303
*150. Internal transformation of a relation	306
*151. Ordinal similarity	319
*152. Definition and elementary properties of relation-numbers	330
*153. The relation-numbers , and	334
*154. Relation-numbers of assigned types	339
*155. Homogeneous relation-numbers	344
SECTION B. ADDITION OF RELATIONS, AND THE PRODUCT OF TWO RELATIONS	347
*160. The sum of two relations	351
*161. Addition of a term to a relation	357
*162. The sum of the relations of a field	362
*163. Relations of mutually exclusive relations	369
*164. Double likeness	376
*165. Relations of relations of couples	386
*166. The product of two relations	396
SECTION C. THE PRINCIPLE OF FIRST DIFFERENCES, AND THE MULTIPLICATION AND EXPONENTIATION OF RELATIONS	403
*170. On the relation of first differences among the sub-classes of a given class	411
*171. The principle of first differences (continued)	423
*172. The product of the relations of a field	428
*173. The product of the relations of a field (continued)	443
*174. The associative law of relational multiplication	447
*176. Exponentiation	458
*177. Propositions connecting with products and powers	471
SECTION D. ARITHMETIC OF RELATION-NUMBERS	473
*180. The sum of two relation-numbers	477
*181. On the addition of unity to a relation-number	482
*182. On separated relations	487
*183. The sum of the relation-numbers of a field	496
*184. The product of two relation-numbers	501
*185. The product of the relation-numbers of a field	505
*186. Powers of relation-numbers	507
PART V. SERIES.
Summary of Part V.	513
SECTION A. GENERAL THEORY OF SERIES	516
*200. Relations contained in diversity	518
*201. Transitive relations	525
*202. Connected relations	533
*204. Elementary properties of series	547
*205. Maximum and minimum points	559
*206. Sequent points	577
*207. Limits	594
*208. The correlation of series	605
SECTION B. ON SECTIONS, SEGMENTS, STRETCHES, AND DERIVATIVES	612
*210. On series of classes generated by the relation of inclusion	615
*211. On sections and segments	624
*212. The series of segments	651
*213. Sectional relations	668
*214. Dedekindian relations	684
*215. Stretches	691
*216. Derivatives	700
*217. On segments of sums and converses	710
SECTION C. ON CONVERGENCE, AND THE LIMITS OF FUNCTIONS	715
*230. On convergents	720
*231. Limiting sections and ultimate oscillation of a function	727
*232. On the oscillation of a function as the argument approaches a given limit	737
*233. On the limits of functions	745
*234. Continuity of functions	753

ADDITIONAL ERRATA TO VOLUME I.

p. 5, line 20, delete "."
p. 34, line 20, for "" read "."
p. 36, line 7 and line 10, for "" read "."
p. 44, line 17, for "" read "."
p. 112, in *2·52, in place of "" read "."
p. 129, in *5·11, in place of reference to "" read reference to "."
p. 129, in *5·12, in place of reference to "" read reference to "*2·51."
p. 144, *10·23 should be "."
p. 157, line 11, for "*10" read "*9."
p. 184, last line of Dem. of *14·111, for second "" read "."
p. 228, in *23·81, for "" read "."
p. 242, in *25·37, for "" read "."
p. 242, in *25·412, for "" read "."
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 "" read "."
p. 289, 1st line, for "" read "."
p. 322, in *40·18, enunciation, for "" read "".
p. 329, in *40·69, Dem., for "" read "" (3 times).
p. 387, in *55·224, 1st line of Dem., for "" read "" (twice).
p. 388, in *55·281, for third "" read "."
p. 410, in *60·53, last line of Dem., for "" read "."
p. 453, in *71·25, Dem., 1st line, for "" read "."
"        "                              2nd line, for ""
read "."
"        "                              3rd line, for "" read "."
"        "                              6th line, for "" read "" and for
"" read "."
"        "                              7th line, for "" read "."
p. 465, in *72·16, Dem., 1st line, for last "" read "."
p. 483, in *73·44, Dem., 1st line, for second "" read "."
p. 485, in *73·511, for "" read "."
p. 522, in *81·23, enunciation and 2nd line of Dem., for "" read "."
p. 592, in *91·33, Dem., 1st line, for "" read "."
p. 614, in *93·36, Dem., for "" read "" throughout.
p. 628, in *95·21, Dem., line 6, for "" read "."

ERRATA TO VOLUME II

p. 82, last line but one, for "" and "" read "" and "."
p. 101, *112·23, enunciation, the second time two dots occur, read one dot.
p. 573, *205·7, enunciation, for "" read "."

PREFATORY STATEMENT OF SYMBOLIC CONVENTIONS — FOOTNOTES:

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 , where "" 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 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 "," the must be of some definite type, i.e. we only assert that is true whatever may be within some one type. Thus e.g. "" does not assert more than that this assertion holds for any 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 "," where is an elementary proposition, we may substitute for "," where is any matrix. Thus instead of "," which contained one variable of a given type, we have "," 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 contains as a constituent, and therefore is not a constituent of even if is a value of . Thus we pass, above, from an assertion containing no function as a constituent to one containing one or more functions as constituents. The assertion "" concerns any elementary proposition, whereas "" concerns any of a certain set of elementary propositions, namely any of those that are values of . Different types of functions give different sorts of ways of picking out elementary propositions.

Having assumed or proved "," where is elementary and therefore involves no ambiguity of type, we thus assert 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 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 , where is individual. Thus we need not assume that elementary propositions form a type; we may replace by "" in "." 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 "" and "," we have "" which is practically *9·12. This was asserted in *1·11 for any case in which and 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 where replaces the of *1·11, and , replace and . 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 and 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 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—, , !, ()., ().—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 individuals, but any other starting-point is equally legitimate. Whatever type of functions (including 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 . Then if is any predicative function of individuals, . 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 without having .

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 and . 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, , then this proposition is false for any higher type; but this proposition, , 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 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, . Then , are two different functions, which, by the axiom of reducibility, are equivalent to two different predicative functions. Hence there are at least two predicative functions of , and at least two classes , . 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 as a primitive idea, put Then consider the proposition . We can prove Thus 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 , and by . Then we have and this holds whatever the type of 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 and ), 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 ) for the type (where ); then merely writing for , we have a proposition which is proved concerning , namely , and here, by what was said before, ) may be replaced by any higher type.

Thus given a typically ambiguous relation , such that, if is a type, is a type ( or is such a relation), we "see" that, if we can prove , we can also prove , where is any type, and is composed of typically ambiguous symbols. Similarly if we can prove , we can prove , where is any type. But we cannot in general prove or , and these may be in fact untrue. e.g. we have .

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

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.

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 , or of the symbol , which is . 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 is true, and in other types may not be true. The fallacy consists in neglecting this latter possibility of the failure of for a limited number of types, that is, in taking the "fluctuating" form 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 . 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 cannot be proved. But 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.

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 such that, in whatever type the constant symbol is determined, it is, in that type, identical with . In other words if be a constant symbol, then is a formal number provided that "truth" is the permanent truth-value of , for some constant .

The functional formal numbers are defined by enumeration; they are where in each formal number the symbols , , , occurring in it are called the arguments of the functional form even when they are complex symbols. The argument of is , and those of are and , and those of are 1 and 2.

Thus among the constant formal numbers are 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 It will be observed that e.g. 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 and , are members of 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 is an argument of , and of ( and of (; also is an argument of (. 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 are occurrences of the same formal number, even if they are determined to be in different types; but and are different formal numbers. Also and are different formal numbers because their "forms" are different, though the arguments 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 , , , (ii) the argumental set consisting only of , (iii) the arithmetical set consisting of , , , and .

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 , , , and are components of (; and , and are components of ; and , and are components of . The two arguments of ( are and , and those of are and , and those of are and .

Addition, multiplication, exponentiation, and subtraction will be called the arithmetical operations; and in , , , , and 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 , , , are arithmetical components of ; and and are arithmetical components of , but is not one; and and are arithmetical components of , but is not one; and and are arithmetical components of , but and and are components of and are therefore not arithmetical components of . 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 and are pure, but and are not pure.

There are many types involved in the consideration of a formal number. For example, in there is the type of and of ; in there is the type of , the type of , and the type of ; 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 by writing , , (, (, (, by the notation of *65. Even when the actual type of a complex formal number, such as , is settled—so for instance that we have —the meaning of the symbol is not completely determined, for the type of 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.

But in order to state this convention 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 , the actual type of should be high enough to enable us logically to prove 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 by another one so related to that with the same actual type for both we can prove , whenever is not equal to in all types. If 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 has the required property. Thus:

(i) The actual types of , , , and are adequate when we can logically prove

(ii) The actual types of , , , and are adequate when we can logically prove

It will be noticed that , , and are the greatest classes of the same type as , , and respectively, and that and are the greatest cardinal numbers of the same type as and respectively. These definitions hold even when any of , , , are complex symbols.

The remaining formal numbers which are not functional must certainly be constant. The difficulty which arises here is that if be such a formal number and occurs in its symbolism, we have no logical method of deciding as to the truth or falsehood of in any type. But we replace by which is the greatest existent cardinal of the same type as in that occurrence. Thus:

(iii) If be a formal number which is not functional, an adequate actual type of is one for which we can logically prove , where is derived from by replacing any occurrence of in by . Accordingly if does not occur in , an adequate type is any actual type for which we can logically prove .

In the case of members of the primary and argumental groups we have substituted the of the appropriate type in the place of each variable. When the actual type is adequate we have

In the case of members of the arithmetical group (except in the case of , we have substituted for each argument the largest cardinal number which can be obtained in the type of that argument, namely the for the of the appropriate type. Accordingly we are sure (except in the case of 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 the actual type of is adequate, and *110·23 shows that any adequate actual type of will do. But nothing is said about the actual type of . 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 for any values of their variables.

Also in the case of those variable formal numbers which are of the pure arithmetical set (excluding ) 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 they can be reduced to the case of the formal numbers of the primary group; and that therefore they are not equal to when their types are arithmetically adjusted. For example in , , , , are these ultimate components; let them be existent cardinal numbers. Hence when the types are arithmetically adjusted, the actual type of is adequate and is an existent cardinal; we can therefore substitute for it. By the same reasoning we can substitute for , and again for .

A definite standard arithmetical adjustment of types for any formal number can always be found by making every use of , whether explicit or concealed in 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] is called an arithmetical inequality.

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, and , or any symbols replacing and , are called the opposed sides of the inequality, and either of or is called a side of the inequality.

Symbolic forms of the kinds and , where either or is a formal number, will be called equations and inequations respectively; and and 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 can occur in a symbolic form are named as follows:

The occurrence of in is called an argumental occurrence,

The occurrence of 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 as one side of an equation is called an equational occurrence,

The occurrence of in "" is called an attributive occurrence,

Any other occurrence of is called a logical occurrence, so also is .

It is obvious that a pair of opposed sides of an equation or inequation must be of the same type. Furthermore, if be a formal number, and *20·18 is applied so as to give the equational occurrence of must be of the same type as its occurrence in ), 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.

Here the formal numbers are and , each of which has three occurrences. The first occurrence of is logical, its second is equational, and its third is attributive.

*100·42 (in the demonstration).

Here and are the only formal numbers, and all their occurrences are equational.

*100·44 (in the demonstration).

Here and are the only formal numbers; the first occurrence of is logical, its second is equational; both the occurrences of are equational.

*100·511.

Here the formal numbers are and . The first occurrence of is logical, the second is argumental, the third is equational; the only occurrence of is equational.

*100·521.

Here and are the only formal numbers; has two occurrences, the first logical, the second argumental; has one occurrence, which is equational.

*101·28 (in the demonstration).

Here the formal numbers are 1 and . The first occurrence of 1 is argumental, the second is attributive; the occurrence of is attributive.

*101·38.

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

*110·54.

Here the formal numbers are The occurrence of ) and that of ( 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 are as arithmetical components of a more complex arithmetical formal number and are therefore arithmetical.

*116·63.

The formal numbers are , , , and (. Each formal number occurs once only. The occurrences of and are arithmetical, and those of the other two are equational.

*117·108.

The formal numbers are and , 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).

Here the formal numbers are , , , , . Each formal number has one occurrence. Those of , and are equational, and those of and are arithmetical.

*120·53 (in the demonstration).

Here the formal numbers are , , , , . The first occurrence of is equational, its second occurrence is logical; the first two occurrences of are equational, its third occurrence is arithmetical; the only occurrence of is arithmetical; the only occurrences of and of are equational.

IV. The Conventions and .

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.

. 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, does not in any way apply to the assignment of types to the occurrences in the form of that formal number.

The identification of types in argumental and attributive occurrences by is rendered necessary to secure the use of the equivalence where 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, directs the logical and equational occurrences of to be in the same type, and similarly for . Also "meaning" secures that the equational types of and 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 and . Thus, using the notation of *65·04 to secure typical definiteness, *100·35 is to mean

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, produces no limitation of types.

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

In *100·511, directs that the logical and argumental occurrences are to have the same type. In *100·521, directs that the two occurrences of are to have the same type. In *101·28 both occurrences of 1 are to be in the same type. In *101·38, directs that all the occurrences of 2 are to have the same type.

The convention 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 has no application.

In the second example from *120·53 (in the demonstration) convention directs that the two occurrences of shall be in the same type; and the necessity of "meaning" secures that the first occurrence of shall also be in this type. The same necessity secures that shall be in the same type as ; and it also secures that in "" the first occurrence of and that of shall have a common type, which is otherwise unfettered; also nothing has been decided as to the types of and in .

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. , , , 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:

. Whenever a formal number occurs, so that, if it were replaced by , the actual type of would by definition have to be adequate, then the actual type of is also to be adequate.

For example in , if were replaced by , then by *110·04 the actual type of is adequate. Hence by the actual type of is to be adequate: accordingly so long as and are simple variables and members of , we can always assume for the type of the occurrence of .

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 *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 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, is a pure arithmetical formal number when , , are determined in type; and convention directs that the type of is to be adequate. But is an arithmetical formal number which is not pure, and convention directs that the type of the domain of is to be adequate, but does not affect the type of . Thus it is easy to see that 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 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 and occur in the same form, then these two occurrences of 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 .

The meaning of 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 and adequately when forming , also to determine and adequately when forming . The convention does not apply to the types of and . These types must be identical in order to secure meaning.

In *116·63 the convention directs us to adjust the types of and adequately; it does not affect the types of and , which must be identical to secure meaning. If we replace , , by formal numbers, by 2, , and 1 for example, we get "." The convention now directs that 1 is to be determined adequately. It so happens that any type is adequate for it, since can be proved in any type. Then adequate types for and are types for which we can prove and . Thus if is the type of in both cases, an adequate type for is , and for is .

In *117·108 we find arithmetical occurrences in arithmetical inequalities. Thus directs us to take the first two occurrences of and the first two of with adequate actual types. The type of and in is not affected by it. It is evident that the conventions , 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 needs a consideration of the whole question of arithmetical substitution. Consider the first of the two examples. We have

It is obvious that unless we can pass with practical immediateness from "" to "" by *20·18, arithmetic is made practically impossible by the theory of types. But a difficulty arises from the application of . Suppose we assign the types of our real variables first. Then the types of , , , can be arbitrarily assigned, and there is no necessary connection between them which arises from the preservation of meaning. Thus may be in a type which is not an adequate type for . Assume that this is the case. But the equational use of is in the same type as , and by the arithmetical use of in 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 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 "," where and are formal numbers and occurs arithmetically in , we cannot infer unless the equation is arithmetical. For otherwise the in the equation cannot be identified with the in .

When we have "," where is a formal number and is a number in a definite type, and wish to pass to "," or "" and wish to pass to "," the occurrence of in being arithmetical, the type of may not be an adequate type for . Accordingly the in "" cannot be identified with the in . The type of the in the equation ought to be freed from dependence on that of . Accordingly the transition is only legitimate when we can write instead 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 . But is easy to see that, the occurrence of being arithmetical, we always have In order to prove this, we have only to prove

The demonstration of the first of these propositions runs as follows:

In the above demonstration the step to (3) is legitimate since by the hypothesis is a determination of 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 "" to "," where the latter equation is arithmetical. In other words, using *65·01 we require the hypothesis necessary for

We have

Now in (4) the occurrences of and , which are in the same type, may be chosen to be in any type we like. Hence we deduce

Hence is the requisite condition. Now since can be in any type, we can also choose it in any existential type for . Thus with applying to the arithmetical occurrence of in , we have, where is a formal number and is a number in a definite type,

In the last proposition by the equation is arithmetical. These equations are summed up in *118·01.

These three fundamental theorems embody the principle of arithmetical substitution. The hypothesis is really less than is assumed in ordinary life, the usual tacit assumption being . In fact unless , is necessarily false.

Principle of Identification of Types. Suppose we have proved "" and "," where is a formal number whose occurrence in "" is in an entirely ambiguous type, and is the same formal number with its type related to that of by *65·01. Then since the type of the in "" is ambiguous, we can write "" and thence infer "."

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 occurs, the first occurrences of and are of the same type, and so are their second occurrences in . But the two types are not determined by our conventions to have any necessary connection. In fact the type in is entirely arbitrary. Accordingly it can be identified with the other type, and thus the inference to the next line, viz. to "," is justified.

In the case of arithmetical equations, it is important to notice that we have Hence if and are formal numbers, Thus if we have "" and "," we can infer from the former proposition "" and from this and the latter proposition, we infer "," so the general principle of identification can be employed when the in the first proposition is an arithmetical equation.

For example, in an example given above, *100·44 (demonstration), viz. the equation is arithmetical. Accordingly we are justified in asserting the propositional function where in "" has all along been presupposed by the necessity of meaning. Thus the inference follows,

This proof loses its point when is looked on as a variable with necessarily the same type throughout. For then the proposition collapses into

But if be a formal number necessarily a member of , the proposition is really

With this presupposition we should have in the first line of the demonstration though with "" 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 can be recognized as a particular case of , where is a real variable and is a formal number. In the first place obviously we must substitute for , wherever it occurs in , and thus obtain . Then we may find that by the application of our conventions, we can replace this by . For example we have

*100·42.

Now put for , we obtain

Now by , even when is a formal number, the identity of types of the two occurrences of is equally secured in

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

Ambiguity of . It follows (cf. *100·02 and *103·02) from the typical ambiguity of that is also typically ambiguous. Hence "" according to our methods of interpretation would not necessitate that and should be of the same type. We shall always interpret "" as standing for "" and therefore as not necessarily identifying the types of and . Similarly for , , and . For example

*110·402.

Here the and need not be of the same type. Again

*110·41.

Here the identification of the types of and requires the hypothesis "."

VI. Conventions and .

General Arithmetical Convention. Conventions and 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 .

. 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 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 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 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.

If we apply to this, we can write

For the equational occurrences of and are by and to be with adequate actual types. But if is a small class in a high type, an adequate actual type for will be a high type, whereas may hold in a low type. Thus with , 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 , though it would unduly limit the scope of *101·34. In *110, 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. is already applied to *110·71·72; if is removed from these propositions, then must be added as an hypothesis to both of them. The effect of on *113 and *116 is entirely analogous to that on *110; in neither of these two numbers is there any proposition to which is applied in the text.

As regards *117, 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, 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 are *117·21·54·592. Propositions involving inequations whose enunciations are unaltered by the removal of are *117·26·27. Finally the only propositions of *117 whose enunciations are altered by the removal of are *117·108·211·23·25.

In *118 and *119 is not used.

In *120, which is devoted to those properties of inductive cardinals which are of logical interest, 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 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 "," is required in certain propositions where the hypothesis implies that there are types in which every inductive cardinal exists, i.e. in which is not an inductive class. Among such hypotheses are , , (or typically definite forms of these hypotheses), or or . When such hypotheses occur, we shall assume that 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:

. When the hypothesis of a proposition implies that there is a type in which every inductive cardinal exists, every occurrence of "" 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 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 It will be seen that these definitions do not suffice to determine the type of . Hence in (1), the on the left may not be of the same type as the on the right. Now the use of *122·473, which occurs in the next line of the demonstration of *122·33, requires that the on the left and the on the right should be of the same type. This requires that the should not be taken in a type in which we have . Hence in order to apply *120·473, we must choose a type in which all inductive cardinals exist. Since "" occurs in the hypothesis, we know that all inductive cardinals exist in the type of . But it is unnecessary to restrict ourselves to the type of , since any other type in which all inductive cardinals exist will equally secure the validity of the demonstration. Thus the convention secures the restriction required, and no more.

The convention is often relevant when "" without any typical determination occurs in the hypothesis. Whenever this is the case, if "" 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 is used, together with 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 (, 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 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 .

We make the definition

*126·01.

Wherever this symbol "" for the class of "indefinite inductive cardinal numbers" is used, the above rule is adhered to. In other words, "" can always be replaced by "," where is a homogeneous or ascending cardinal, and is the appropriate constant, or is a variable, as the case may be. In the latter case, a symbolic form such as can be replaced by

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 and ; for , , , , , , and , and so on.

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

PART III. CARDINAL ARITHMETIC.

SECTION A. DEFINITION AND LOGICAL PROPERTIES OF CARDINAL NUMBERS. — FOOTNOTES:

Summary of Section A.

The Cardinal Number of a class , which we will denote by "," is defined as the class of all classes similar to , i.e. as . 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 is any object, 1 is not the cardinal number of , but that of . This obviates a confusion which otherwise is liable to arise in dealing with classes. Suppose we have a class 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 that is many, and 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 to some simplification. It will be seen that, according to this formula, "" 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 the relation ; so has 2 to , provided . The relation is, in fact, the relation ; for . Hence for formal purposes of definition we put

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 , are equal to . We call the class of cardinal numbers ; thus we have

For purposes of formal definition, we replace this by the simpler formula

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 "" is typically ambiguous; the only limitation on its type is that its domain and converse domain must both consist of classes. When we have , and 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 , whatever types and may belong to. This ambiguity of "" is derived from that of , which in turn is derived from that of 1. We denote (cf. *65·01) by "" all the unit classes which are of the same type as . Then (according to the definition *70·01) will be the class of those one-one relations whose domain is of the same type as and whose converse domain is of the same type as . Thus "" is typically definite as soon as and are given. Suppose now, instead of having merely , we have then we know not only that , but also that belongs to the same type as , and belongs to the same type as . When the ambiguous symbol "" is rendered typically definite by having its domain defined as being of the same type as , and its converse domain defined as being of the same type as , we write it "," because generally, in accordance with *65·1, if is a typically ambiguous relation, we write for the typically definite relation that results when the domain of is to consist of terms of the same type as , and the converse domain is to consist of terms of the same type as . Thus we have Here everything is typically definite if and (or their types) are given.

Passing now to the relation "," it will be seen that it shares the typical ambiguity of "." In order to render it typically definite, we must derive it from a typically definite "." So long as nothing is added to give typical definiteness, "" will mean all the classes belonging to some one (unspecified) type and similar to . If is a member of the type to which these classes are to belong, then is contained in the type of . For this case, it is convenient to introduce the following two notations, already defined in *65. When a typically ambiguous relation 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 , we write "" in place of . When we further wish to determine as having members of the converse domain contained in the type of , we write "" in place of ; and when we wish members of the converse domain to be members of the type of , we write "" in place of . Thus (cf. *65·2), and in particular, since , Thus "" is only significant when is of the same type as , and then it means "classes of the same type as and similar to (which is of the same type as )."

"" will mean "classes of the same type as and similar to ." As soon as the types of and are known, this is a typically definite symbol, being in fact equal to . Hence so long as we only wish to consider "," typical definiteness is secured by writing "" in place of "."

When we come to the consideration of , "" is no longer a sufficient determination, although it suffices to determine the type. Suppose we put we have also, in virtue of the definitions in *65, Thus 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 whose truth or falsehood depends upon the determination chosen for the converse domain of . Hence if we wish to have a symbol which is completely definite, we must write "."

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 is any class, no class contained in is similar to . Hence in particular if is a type, no class contained in is similar to , which is the next type above . Consequently, if , where is any class, we have

Now (cf. *63) we put and we have . Thus we find That is to say, no class of the same type as has as many members as has. Hence also and "" is only significant when ; hence

Now the notation "" will apply with equal justice to or to ; but we have just seen that in the first case we shall have , and in the second we shall have . Consequently "" 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 be a class of classes; then is of lower type than . Let us consider . In accordance with *63, we write for the type containing , i.e. for . Then the greatest number in the class will be ; but neither this nor any lesser member of the class will be equal to , because, as before, Hence , which is a member of , is not a member of ; but and have an equal right to be called . Hence again "" 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 , 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 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 . 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 is But if we take the corresponding cardinal in the next type, i.e. this is not as great as , 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 and , as above, but merely what we may call their relative types. Thus, for example, and may be of the same type; in that case, and are respectively equal to and . We will call cardinals which, for some , are members of the class , homogeneous cardinals, because the "" from which they are derived is a homogeneous relation. We shall denote the homogeneous cardinal of by "," and we shall denote the class of homogeneous cardinals (in an unspecified type) by ""; thus we put Almost all the properties of are the same in different types. When further typical definiteness is required, it can be secured by writing , in place of , . For although and were not wholly definite, and are wholly definite. Apart from the fact of being of different types, the only property in which and differ when and 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 "." Then will consist of 0 and 1, i.e. But in the next higher type, there will be two members, namely and . Thus the members of being , , ; and so on. (The greatest cardinal in any except the lowest type is always a power of 2.)

The maximum of is ; but apart from this difference of maximum and its consequences, and do not differ in any important properties. Hence for most purposes and 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 is called an ascending cardinal if the type of is or or or etc. We write for , for , and so on. We put We then have obviously We also have (by what was said earlier) The members of will be all cardinals which exceed but do not exceed .

Let us recur in illustration to our previous hypothesis of the universe consisting of a single individual. Then will consist of those classes which are similar to "" but of the next higher type. These are and . In our case we had . This leads to or, introducing typical definiteness, We have then . Also And in the case supposed, is the maximum of , but . Hence Generalizing, we see that consists of the same numbers as each raised one degree in type. Similar propositions hold of , etc.

It is often useful to have a notation for what we may call "the same cardinal in another type." Suppose is a typically definite cardinal; then we will denote by the same cardinal in the next type, i.e. Note that, if is a cardinal, ; and whether is a typically definite cardinal or not, is a cardinal in a definite type. If is typically definite, then is wholly definite; if is typically ambiguous, has the same kind of indefiniteness as belongs to . The most important case is when is typically definite and has an assigned relation of type to . We then put, as observed above, If is an , is an and is an and so on. will consist of all numbers which are of the form for some which is a member of ; i.e.

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. is a descending cardinal if is of a lower type than . We put

We have obviously .

Hence .

Also , whence , whence .

Since also , we find 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 is similar to , we have Consequently .

We can prove in just the same way 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 . (We shall take this as the definition of the product of the numbers of the members of .) Suppose now that consists of a single term: we want to be able to say We have in this case, if , and we know that . But if we put simply our proposition, though not mistaken, requires care in interpretation. Just as we put , so we want a notation giving typical definiteness to the proposition . This is provided as follows.

Using the notation of *64, put Then we have, for example, Hence , where .

Similarly .

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 , or of any relation whose domain and converse domain have respectively given relations of type to the domain and converse domain of . Thus we might put ("b" appears here as "d" written backwards)

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 "" or "," 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 "," although, if and are of different types, the two symbols "" 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 because here "" must mean "," so that the only ambiguity remaining is as to the type of , and the formula holds whatever type may belong to, provided "" is significant, i.e. provided is a class. But we must not, from "," allow ourselves to infer For here the conditions of significance no longer demand that "" should mean "": it might just as well mean "." And as we saw, if is a lower type than , and is sufficiently large of its type, we may have so that "" 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 or .

*100. DEFINITION AND ELEMENTARY PROPERTIES OF CARDINAL NUMBERS.

*101. ON 0 AND 1 AND 2.

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 and exist; this follows from and . We prove (*101·22·34) that 0 and 1 and 2 are all different from each other. We prove (*101·15·28) that and , but we cannot prove unless we assume the existence of at least two individuals, or define the first 2 in "" as a 2 of some type other than , where "" 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 "" rather than to those of , where . For example, we have

*100·511.

but we shall not have unless the "" 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) where 0 and 1 and 2 have an ambiguity corresponding to that of "."

*101·1.

*101·11.

*101·12.

*101·13.

*101·14.

Dem.

*101·15.

Dem.

*101·16.

Dem.

*101·17.

Dem.

*101·2.

*101·21.

*101·22.

Dem.

*101·23.

Dem.

*101·24.

Dem.

*101·241.

*101·25.

Dem.

*101·26.

Dem.

*101·27.

Dem.

*101·28.

Dem.

*101·29.

Dem.

*101·3.

Dem.

*101·301.

In comparing *101·31 with *101·1·2·3, it should be observed that and 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.

Dem.

*101·32.

*101·33.

*101·34.

Dem.

*101·35.

*101·36.

Dem.

*101·37.

*101·38.

Dem.

*101·4.

Dem.

When we are considering the lowest type occurring in a context, our premisses do not suffice to prove . For every other type, this can be proved. Thus and give the required result for classes and relations respectively.

*101·41.

Dem.

*101·42.

Dem.

*101·43.

*102. ON CARDINAL NUMBERS OF ASSIGNED TYPES.

Summary of *102.

In this number, we shall consider a typically definite relation "," i.e. we shall consider the relation, to a class which is given as of the same type as , of the class of those classes which are similar to and of the same type as . We shall then put and the class of all such numbers as for a given and we shall call , so that

The notations here introduced for giving typical definiteness to "" and "" are those defined in *65 for any typically ambiguous relation.

By *63·01·02 we have, if is a typically ambiguous symbol,

Thus . If we apply the definitions to 1, "" is meaningless unless is a class; we therefore write a Greek letter in place of , and we have

If , we shall have . Hence

The converse implication also holds, so that

Thus consists of all unit classes whose sole members either are or are not members of , i.e. for which "" is significant.

In "," the hypothesis renders explicit the condition of significance; thus "" is always true when significant, and always significant when . 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. , , , , , , , , , . These 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 , it is not true that, conversely, when the type of the referent is given, that of the relatum is also given.) But "" and "" 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 "" or "" must be a class, that the referent for "" must be a class, and that the referent for "" must be a class of classes. When a relation has relative definiteness, it is enough to fix the type of the relatum; and if further , so that leads to a descriptive function, "" has complete typical definiteness as soon as the type of is given. Now the constant relations hitherto introduced, with the exception of "" and "," 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 "" in these circumstances, has typical definiteness as soon as is assigned. But with the consideration of "" and "," which do not have even relative definiteness, an explicit means of giving typical definiteness becomes necessary. It should be observed, however, that "" has typical definiteness, when is known, as soon as the domain of "" has typical definiteness, since 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 is a typically ambiguous relation, and is a referent, becomes ; if, further, is a relatum, becomes . If is a referent for , we have , and . Thus has a member of the type next above that of , i.e. of the type of . Thus as was proved in *65. Hence in particular

It is chiefly for this reason that it is worth while to introduce the definition of .

We have, in virtue of the above, as will be proved in *102·46,

With regard to "," which is to be interpreted by *65·04, some caution is necessary. This will mean some one of those typically different relations called "" which have their domains composed of terms of the same type as . But it will not mean the logical sum of all such relations, because these relations are of different types according as their converse domains differ in type, and therefore their logical sum is meaningless. Thus for example if the type of is lower than or equal to that of , we shall have whence, if "" has its converse domain composed of terms of the same type as , But if is of higher type than , we shall find Thus "" is indeterminate in a way that makes a practical difference.

Exactly similar remarks apply to . We have thus "" shares the ambiguity of "." The question whether depends upon the decision of this ambiguity. The difficulty is that "" stands for the domain of any one determination of "" which has its domain composed of objects of the type of ; but it is the domain of only one such determination of "," 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, "" is a symbol which is as a rule better avoided, and "" is not often useful except as a descriptive function, in which case the relatum supplies the requisite typical definiteness.

The peculiarity of "" 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 "" half-definite, as "" does, but must make it completely definite, as we do by taking . For this we adopt the notation . We cannot adopt the notation , because that would conflict with *65·11, nor , because that would conflict with *65·01, nor , for the same reason. But has no previously defined meaning. We may if we like regard "" as . Then the required meaning of "" would result from *65·04. But as "" so defined is not required, it is simpler to regard "" as a single symbol. We therefore put

*102·01.

The present number begins with various propositions (*102·2—·27) on a typically definite relation of similarity, i.e. . We then have a set of propositions (*102·3-·46) on "." This is only significant if and are of the same type; it then denotes the class of those classes which are similar to and of the same type as . We then have a set of propositions (*102·5-·64) on , i.e. on cardinals consisting of classes of the same type as which are similar to classes of the same type as . We next prove (*102·71—·75) that no sub-class of \alpha is similar to , and therefore (substituting for ) no class of the same type as is similar to , and therefore

*102*74.

This proves that is a cardinal, which is a proposition constantly required. The remaining propositions of *102 are concerned with where is a typically definite cardinal.

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

*102·3.

*102·46.

*102·5.

*102·6.

*102·72.

This is used in proving , which is the proposition from which Cantor deduced that there is no greatest cardinal. (If , , and thus there is a rise of type.)

*102·84.

*102·85.

*102·01.

*102·11.

Here, if is a real variable, the conditions of significance require . But if is a typically ambiguous constant, such as or or , is a typically definite constant. It is chiefly for such cases that propositions such as the above are useful.

Dem.

*102·13.

*102·2.

*102·21.

*102·22.

102·23.

*102·24.

Dem.

*102·25.

*102·26.

Dem.

*102·27.

*102·3.

Dem.

*102·31.

Dem.

*102·32.

Dem.

*102·34.

*102·35.

*102·36.

This proposition is true whenever it is significant, and is significant whenever . When belongs to some other type, the above proposition is not significant.

*102·361.

*102·37.

Dem.

*102·4.

*102·41.

*102·42.

*102·43.

This inference is legitimate because, when is given, "" is typically definite. The inference from "" (which is true) to "" is not valid, because "" may hold only for some of the possible determinations of the ambiguity of "."

*102·44.

Dem.

*102·45.

Dem.

*102·46.

*102·5.

In using propositions, such as those of *100, in which we have a typically ambiguous "" or "," 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.

*102·51.

Dem.

*102·52.

*102·53.

Dem.

*102·54.

*102·541.

Dem.

*102·55.

Dem.

The above proposition shows that, if every class of the same type as is similar to some class of the same type as , then, given a class of the same type as , there is a class , of the same type as , such that the classes similar to and of the same type as are the same as the classes similar to and of the same type as ; and conversely, given any class , of the same type as , and similar to some class of the same type as , then there is a class , of the same type as , such that the classes similar to and of the same type as are the same as the classes similar to and of the same type as . We may express this by saying that, if the cardinals which go from the type of to the type of are never null, then those that go from the type of to the type of , with the exception of (if is one of them), are the same as those that begin and end within the type of . 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.

Dem.

*102·61.

Dem.

*102·62.

Dem.

*102·63.

Dem.

*102·64.

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 is a cardinal, namely what we call a "descending" cardinal, i.e. one whose corresponding "" goes from a higher to a lower type.

*102·71.

Dem.

*102·72.

Dem.

*102·73.

Dem.

This proposition proves that no class of the same type as is similar to . Now is the greatest class of its type; thus there are classes of the type next above that of which are too great to be similar to any class of the type of . 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.

Dem.

*102·75.

Dem.

*102·8.

Dem.

*102·81.

Dem.

*102·82.

*102·83.

Dem.

*102·84.

Dem.

*102·85.

*102·86.

Dem.

*102·861.

Dem.

*102·862.

Dem.

*102·863.

Dem.

*102·87.

*102·88.

Dem.

Note on negative statements concerning types. Statements such as "" or "" 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, and 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 "" 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 "" from the (true) propositions "" and "." (The truth of the first of these two requires that "" 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.

*103. HOMOGENEOUS CARDINALS.

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 and of the same type as . The "homogeneous cardinal of " will be defined as ; we shall denote it by "." Then the class of homogeneous cardinals is the class of all such cardinals as "," i.e. it is ; this we shall denote by "." The symbol "" is typically definite as soon as is assigned; "," on the contrary, is typically ambiguous: it must be a , but otherwise its type may vary indefinitely. Homogeneous cardinals have, however, many properties which do not require that the ambiguity of "" 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 and , for example, will be defined by means of and , in order to exclude such a determination of the typical ambiguity of and 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.

Other important propositions in this number are the following:

*103·2.

*103·26.

The above proposition is used constantly.

*103·27.

Thus to say that is the homogeneous cardinal of is equivalent to saying that is a cardinal of which is a member.

*103·301.

*103·34.

*103·4.

*103·41.

*103·01.

*103·02.

*103·1.

*103·11.

*103·12.

*103·13.

This is a legitimate inference from *103·12 because, when is given, is typically definite.

*103·14.

Dem.

*103·15.

Dem.

*103·16.

In this proposition, the equation "" must be supposed to hold in any type for which it is significant. Otherwise, we might find a type for which , without having .

Dem.

*103·2.

*103·21.

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

*103·22.

*103·23.

*103·24.

*103·25.

*103·26.

Dem.

*103·27.

Dem.

*103·28.

Dem.

*103·3.

Dem.

*103·301.

Note that although "" is not definite, "" is absolutely definite as soon as is assigned.

Dem.

*103·31.

Dem.

*103·32.

Dem.

In the above proposition, the "" may be omitted, and we may write (cf. *103·33, below)

For the is wholly arbitrary, so that any possible determination of makes the above proposition true. We may proceed a step further, and write (*103·34, below)

But although we also have , provided the "" on the right is suitably determined, we do not have this always. For example, if "" is determined as , and "" as , then .

*103·33.

Dem.

*103·34.

Dem.

Thus every cardinal except is a homogeneous cardinal in the appropriate type. Note that although of course every homogeneous cardinal is a cardinal, yet "" must not be asserted, because it is possible to determine the ambiguity of "" in such a way as to make this false. Hence we do not get .

*103·35.

The hypothesis of this proposition is satisfied, as will appear later, if the type of is in what we may call the direct ascent from that of , i.e. if it can be reached from by a finite number of steps each of which takes us from a type to either or . Thus in such a case the cardinals (other than ) which go from to are the same as those which begin and end within . It will also appear that in such a case always is a member of . If two cardinals which are not equal must always be one greater and the other less, then is the condition for . In that case, we shall have . 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.

Dem.

*103·41.

Dem.

*103·42.

Dem.

*103·43.

Dem.

*103·44.

Dem.

*103.5.

Dem.

*103·51.

Dem.

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) , so that in that type .

*104. ASCENDING CARDINALS.

Summary of *104.

In this number we have to consider cardinals derived from a relation of similarity which goes from the type of to that of , or to that of . The propositions to be proved can be extended, by a mere repetition of the proofs, to , , 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

Ascending cardinals, though less important than homogeneous cardinals, yet have considerable importance in arithmetic, because and are defined as the cardinals of classes of higher types than those of and , 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.

*104·02.

*104·03.

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

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

*104·12.

*104·2.

*104·21.

*104·24.

*104·25.

*104·26.

*104·265.

*104·27.

*104·35.

*104·43.

*104·01.

This defines the cardinal number of in the next type above that of ; thus consists of all classes similar to and of the next type above that of .

*104·011.

Similar definitions are to be assumed for , etc.

*104·02.

, like , is typically ambiguous; but is typically definite.

*104·021.

Similar definitions are to be assumed for , etc.

*104·03.

Here, if is a cardinal, is the same cardinal in the next higher type. For example, if is couples of individuals, is couples of classes of individuals.

*104·031.

Similar definitions are to be assumed for , etc.

*104·1.

*104·101.

*104·102.

*104·11.

*104·111.

*104·112.

*104·12.

Dem.

*104·121.

Dem.

*104·122.

*104·123.

*104·13.

*104·14.

*104·141.

When the hypothesis "" is omitted, this proposition is still true, but with a difference. E.g. let us put Then . Thus . But we still have Thus , but is not the same cardinal as in a higher type, i.e. there are classes whose cardinal in one type is , but whose cardinal in the next higher type is not .

*104·142.

*104·15.

*104·2.

Dem.

*104·201.

Dem.

*104·21.

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. , , etc.

*104·211.

*104·23.

Dem.

*104·231.

Dem.

*104·232.

*104·24.

*104·25.

This proposition holds for each possible determination of the typical ambiguities, i.e. for every we have

*104·251.

*104·252.

*104·26.

Dem.

*104·261.

Dem.

*104·262.

Dem.

*104·263.

Dem.

*104·264.

Dem.

*104·265.

*104·27.

*104·28.

*104·29.

Dem.

*104·3.

Dem.

*104·31.

*104·311.

*104·32.

Dem.

*104·33.

Dem.

*104·34.

Dem.

*104·35.

*104·36.

Dem.

*104·37.

Dem.

The following propositions are concerned with the proof that, given any two cardinals and , of the same type, we can find two mutually exclusive classes one of which has terms while the other has terms. The proof requires that we should raise the types of both and one degree above that in which they were originally given, i.e. that we should turn and into and . 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 were this number. Then unless , a class of individuals will be an existent sub-class of the only class which consists of individuals, and therefore we shall have

But if we consider classes of classes and classes, we shall always be able to find a and a such that

The existence of such a and is important in connection with the arithmetical operations, and is therefore proved here.

*104·4.

Dem.

*104·41

Dem.

This proposition proves the desired conclusions provided , and consists of at least three terms. The following propositions deal with the cases in which this hypothesis is not verified.

*104·411.

Dem.

*104·412.

Dem.

*104·413.

Dem.

*104·42.

*104·43.

Dem.

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

*104·44.

*104·45.

*104·46.

*105. DESCENDING CARDINALS.

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 , we consider in the type of members of (which we shall call ) or in some lower type. Thus e.g. we shall have where "" means "classes similar to but of the next lower type." Similarly and so on. We shall have generally and so on. The chief difference between ascending and descending cardinals is that 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 with similar definitions for and .

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.

*105·251.

*105·26.

Thus or , in any given type, only differs from in that type by the addition of .

*105·3.

*105·322.

*105·34.

*105·35.

*105·38.

*105·01.

We might write which would be equivalent to the above. But we choose the above form for the sake of uniformity. If is any suffix, we put, provided has been defined, and if is any index for which has been defined, we put

Thus for the sake of uniformity it is better, in the above definition *105·01, to write "" rather than "."

*105·011.

*105·02.

*105·021.

*105·03.

*105·031.

*105·1.

*105·101.

*105·11.

*105·111.

*105·12.

*105·121.

*105·13.

*105·131.

*105·14.

Dem.

*105·141.

*105·142.

*105·143.

*105·15.

*105·151.

*105·16.

*105·161.

In what follows, propositions concerning or have proofs exactly analogous to those of the corresponding propositions concerning or .

*105·2.

Dem.

*105·201.

*105·21.

*105·211.

*105·22.

*105·221.

*105·23.

*105·231.

*105·24.

*105·241.

*105·25.

*105·251.

*105·252.

Dem.

*105·26.

Dem.

*105·261.

*105·27.

*105·271.

*105·28.

*105·281.

*105·29.

*105·3.

Dem.

*105·301.

*105·31.

*105·311.

*105·312.

*105·313.

*105·314.

*105·315.

*105·316.

Dem.

*105·317.

*105·32.

Dem.

*105·321.

*105·322.

*105·323.

*105·324.

*105·325.

*105·326.

Dem.

*105·327.

*105·33.

Dem.

*105·331.

*105·34.

*105·341.

*105·342.

Dem.

*105·343.

*105·344.

Dem.

*105·345.

*105·35.

Dem.

*105·351.

*105·352.

*105·353.

*105·354.

*105·355.

*105·356.

*105·357.

*105·36.

Dem.

*105·361.

Dem.

*105·362.

*105·37.

*105·371.

Dem.

*105·372.

*105·38.

Dem.

*105·4.

Dem.

*105·41.

*105·42.

*105·43.

Dem.

*105·44.

Dem.

*106. CARDINALS OF RELATIONAL TYPES.

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 , and has a given relation of type to when is given. Thus we want a notation for 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 , , , our relations are contained in

Thus if has been defined, we shall put with analogous definitions for , and .

Much the most important case is that of . For this case we have

*106·1.

Thus will be the number of a class of relations whose fields are of the same type as , provided this class of relations is similar to . E.g. the number of terms such as , where , will be .

We have

*106·21.

*106·22.

*106·23.

*106·32.

*106·4·41·411.

*106·53.

whence it follows that

*106·54.

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

*106·011.

*106·012.

*106·02.

*106·021.

*106·03.

*106·04.

*106·041.

*106·1.

*106·101.

Similar propositions hold for any other double index for which has been defined.

*106·11.

Similar propositions hold for any other double suffix for which has been defined.

*106·12.

*106·121.

Similar propositions hold for any other index and suffix for which or has been defined.

*106·13.

Similar propositions hold for etc.

*106·14.

*106·141.

Similar propositions hold for , , etc.

*106·2

Dem.

*106·201.

*106·202.

*106·203.

*106·204.

*106·21.

*106·211.

*106·212.

*106·213.

*106·22.

Dem.

The proof requires, in addition to *106·12, its analogue for . Such analogues will be assumed as required.

*106·221.

*106·222.

*106·223.

Other propositions of the same kind as the above may be proved by observing that, if and are indices for which and have been defined, we have of which the proof is direct and simple. Hence, since we always have , we also always have We have in like manner But we do not always have

*106·23.

Dem.

*106·231.

*106·24.

*106·241.

The analogues of the above propositions for other indices or suffixes are similarly proved.

*106·25.

*106·251.

*106·31.

*106·311.

*106·312.

Dem.

*106·32.

Dem.

*106·4.

Dem.

*106·401.

Dem.

*106·402.

Dem.

*106·41.

Dem.

*106·411.

*106·43.

Dem.

*106·44.

The following propositions are analogous to *102·71 ff., and similar remarks apply to them.

*106·5.

Dem.

*106·51.

Dem.

*106·52.

Dem.

*106·53.

*106·54.

Dem.

*106·55.

SECTION B. ADDITION, MULTIPLICATION AND EXPONENTIATION.

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 and are mutually exclusive classes, the sum of their cardinal numbers will be the cardinal number of . But in order that and 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 and , we require to find two classes which are mutually exclusive and are respectively similar to and ; if these two classes are called and , then will be the sum of the cardinal numbers of and . We note that and indicate respectively the 's of the same types as and , and accordingly we take as and the two classes these two classes are always of the same type, always mutually exclusive, and always similar to and respectively. Hence we define

The sum of the cardinal numbers of and will then be the cardinal number of ; hence we may call the arithmetical class-sum of two classes, in contradistinction to , which is the logical sum. It will be noted that , unlike , does not require that and should be of the same type. Also is not identical with , but when , is also , though in a different type. Thus the law of tautology does not hold of the arithmetical class-sum of two classes.

If and are two cardinals of assigned types, we denote their arithmetical sum by . (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 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 terms and terms respectively. Hence we are led to the following definition:

In this definition, various points should be noted. In the first place, it does not require that and should be of the same type; is significant whenever and are classes of classes. Thus it is not necessary for significance that and should be cardinals, though if they are not both cardinals, . If they are both cardinals, we find

Thus in this case .

Hence if neither nor is null, and if has terms and has terms, is a member of . It easily follows that

Hence when and are homogeneous cardinals (i.e. when they are cardinals other than ), their sum is the number of the arithmetical class-sum of any two classes having terms and terms respectively.

A few words are necessary to explain why, in the definition, we put rather than . The reason is this. Suppose either or , say , is . Then, by *102·73, , if is of the appropriate type. Hence if we had put where the ambiguities of type involved in and may be determined as we please, we should have We should also have and so on. Thus 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 in the definition, and obtain the typical ambiguity which we desire by means of the typical ambiguity of the "" in "." 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 of the prefatory statement.

The above definition of is designed for the case in which and are typically definite. But we must be able to speak of "," and this must be a definite cardinal, namely . If we simply write in place of , in the definition of , we find But this will not always have a definite value when the type of is assigned. To take a simple case, write for and for . Then whence we easily obtain If we determine the ambiguity of to be , we find in all types; but if we determine the ambiguity to be , we have and this exists in the type of , if not in lower types. Hence the value of depends upon the determination of the ambiguity of . It is obvious that we want our definition to yield in all types; but in order to insure that this shall hold even when, for some values of , , we must introduce two new definitions, namely This definition is to be applied when "" and "" occur without any determination of type. On the other hand, if we have and , we apply the definition of . We shall find that whenever and both exist, Thus the above definition is only required in order to exclude values of or for which either or is .

The commutative and associative laws of arithmetical addition are easily deduced from the definition of . We shall have because each = . A similar though slightly longer proof shows that

The above definition of 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 , and define the arithmetical sum of the members of . Thus now the classes which are the summands must all be of the same type (since they are all members of ), and no one class can occur more than once, since each member of 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 in addition to the definition now to be given.

If is a class of classes, the sum of the cardinal numbers of the members of 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 . Suppose , are two different members of , and suppose is a member both of and of . Then we wish to count twice over, once as a member of and once as a member of . The simplest way to do this is to form the ordinal couples and , which are not identical except when and are identical. Thus if we take all such ordinal couples, i.e. if we take the class for every which is a member of , we get a class of mutually exclusive classes, namely the classes of the form , where , each of these is similar to the corresponding member of . Hence the logical sum of this class of classes, i.e. has the required number of terms. Now, by *85·601, Hence the class whose logical sum we are taking is . Hence we put may be called the arithmetical sum of , in contradistinction to , which is the logical sum. Thus bears to a relation analogous to that which bears to .

We put further

Thus is the sum of the numbers of members of .

It is to be observed that is not in general a function of . For, if two members of have the same cardinal number, this will only count once in , whereas it counts twice in .

We shall find that, provided , 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 and , which is a certain class whose cardinal number is the product of the cardinal numbers of and . We write for the arithmetical class-product of and , and define it as the class of all ordinal couples of which the referent is a member of and the relatum a member of , i.e. as By *40·7, this class is . Hence we put The class is similar to , and each member of it is similar to ; hence if and , consists of classes having members each. The class is important also in connection with exponentiation.

The product of two cardinals is defined as follows: In regard to types, this definition calls for analogous remarks to those which were made on . Also, as before, we need definitions of and , whence we obtain 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 is a class of classes, we take as its arithmetical product. In simple cases, it is easy to see the justification of this decision. E.g. let consist of the three classes , , , and let the members of be , ; those of , , ; those of , , . Then the members of are with four more obtained by substituting for in the above. Thus . In general, however, the existence of 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 is a class of mutually exclusive classes, is similar to . On account of its lower type, is often more convenient than . Hence we put or (what comes to the same thing) For the product of the cardinal numbers of the members of , we put As in the case of , is not in general a function of . We shall have 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 , analogous to or , but only a definition of , which may be extended to any finite number of exponentiations. We put where has the meaning explained above, resulting from *38·03. It will be observed that, if and , is a class of mutually exclusive classes each of which has terms; hence may suitably be used to define . Hence we put and for the same reasons as before, we put

The above definition of exponentiation gives the same value of as results from Cantor's definition by means of "Belegungen." The class of Cantor's "Belegungen" is and it is easily proved that this is similar to .

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 factors, each of which is , is . This proposition cannot be proved generally without the multiplicative axiom. Similarly, in the theory of multiplication, the proposition that the sum of 's is 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: We will take the first of these as being simpler. It affirms that the sum of 's is . This can be proved when is finite, whether is finite or not; but when is infinite, it cannot be proved without the multiplicative axiom. This may be seen as follows. We know that Thus (A) above will result if we can prove since we shall put for , and use (B).

Since , , we have . Assume Let , , ... be members of , and let , , ... be the members of which are correlated with , , ... by , i.e. . We have, since , , . Thus , i.e. . If and are finite, we can pick out arbitrarily a correlation for and , another for and , and so on; then ... correlates and , and therefore . But when and are infinite, this method is impracticable. In this case, we proceed as follows.

By *73·01, .

Thus "" will stand for all the permutations of a class into itself; "" stands for all the permutations of into , i.e. all the 's whose domain is and whose converse domain is . It is obvious that In the case of the and above, we know that when ; thus where "" stands for "correspondence." Thus is the class of all correspondences of and ; 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 and ; i.e. Thus the desired result follows whenever Now we have .

Consequently whence, by what was said previously,

The consideration of leads similarly to the proposition 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 and concerning which we assume i.e. we assume that and are similar classes of similar classes. A slightly modified hypothesis concerning and 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.

Put , where "" is a single symbol representing a relation.

When this relation holds between and , we shall say that and have "double similarity." In this case, correlates and , while correlates and , so that if is a member of , , i.e. , is its correlate in . We shall then have

Also we have Conversely, whence Hence the multiplicative axiom is only required in order to pass from to . It is this fact, and the consequent possibility of diminishing the use of the multiplicative axiom, which has led us to the employment of "" in the present section.

We treat also, in this section, the relation of greater and less between cardinals. We say that when there is a part of which is similar to , but no part of is similar to . The principal proposition in this subject is the Schröder-Bernstein theorem, i.e. 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. 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).

*110. THE ARITHMETICAL SUM OF TWO CLASSES AND OF TWO CARDINALS.

Summary of *110.

In this number, we start from the definition:

*110·01.

is called the "arithmetical class-sum" of and . The definition is framed so as to give two mutually exclusive classes respectively similar to and , so that the number of terms in the logical sum of these two classes is the arithmetical sum of the numbers of terms in and respectively. is significant whenever and are classes, whatever their types may be.

By means of , we define the arithmetical sum of two cardinals as follows:

*110·02.

This defines the "arithmetical sum of two cardinals." (It is not necessary to significance that and should be cardinals, but only that they should be classes of classes. If, however, either is not a cardinal, ). It will be observed that, when and are typically definite, so are and in the above definition; but is typically ambiguous, on account of the ambiguity of "." Hence is also typically ambiguous.

It will be shown that is always a cardinal, and that, if Hence whenever and are cardinals other than , is an existent cardinal in some types, though it may be in others.

Two more definitions are required in this number, namely:

*110·03.

*110·04.

These definitions are needed in order to apply the definition of to the case in which and are replaced by typically ambiguous symbols and . It does not make any difference to the value of how the ambiguities of and are determined, so long as they are determined in a way that insures ; but if there are types in which either or is , we get in all types if we determine the ambiguities so that or . It is in order to 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 of the prefatory statement must be noted.

The propositions of the present number begin with the properties of . We show (*110·11·12) that consists of two mutually exclusive parts, which are respectively similar to and ; we show (*110·14) that if and are mutually exclusive, is similar to , and (*110·15) that if and are respectively similar to and , then is similar to . We show (*110·16) that consists of all classes which can be divided into two mutually exclusive parts which are respectively similar to and .

We then proceed (*110·2—·252) to the consideration of . Here and are typically definite, and the definition *110·02 applies to any typically definite symbols, such as or . We prove (*110·21) that if and are cardinals, their sum consists of all classes similar to some class of the form , where ; we prove (*110·22) that the sum of and is , and (*110·25) that if and 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.

We then (*110·3—·351) consider , to which we apply the definitions *110·03·04. We have

*110·3. whence the other properties of follow from previous propositions.

We then have (*110·4—·44) various propositions on the type of and its existence and kindred matters. The chief of these are

*110·4.

*110·42.

This proposition requires no hypothesis, because, if and are not both cardinals, , and 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 .

*110·01.

*110·02.

*110·03.

*110·04.

These definitions are extended by of the prefatory statement.

*110·1.

*110·101.

Dem.

*110·11.

Dem.

*110·12.

*110·11·12 give the justification for the use of in defining arithmetical addition, since they show that consists of two mutually exclusive parts which are respectively similar to and .

*110·13.

Dem.

*110·14.

Thus whenever and are mutually exclusive, their logical sum may replace their arithmetical sum in defining the sum of their cardinal numbers.

*110·15.

Dem.

*110·151.

Dem.

*110·152.

Dem.

*110·16.

*110·17.

Dem.

Thus when and are of the same type, exists at least in the type next above that of and . We cannot prove that it exists in the type of and . E.g. suppose the lowest type contained only one member; then if were that one member, would not exist in the type to which belongs, but would exist in the next type, i.e. there would not be two individuals, but there would be two classes, namely and , so that .

*110·18.

Dem.

*110·2.

*110·201.

*110·202.

Dem.

*110·21.

*110·211.

Dem.

*110·212.

Dem.

*110·22.

Dem.

*110·221.

Dem.

*110·23.

Thus is independent of and so long as and exist in the types of and respectively.

*110·231.

*110·24.

Dem.

*110·25.

Dem.

*110·251.

Dem.

*110·252.

A similar proof applies to , , etc., and to any such derived cardinals whose existence follows from that of and . The proposition does not hold generally for , and other descending derived cardinals, because they may be null when and exist.

The following proposition (*110·3) is more often used than any other in this number except *110·4.

*110·3.

*110·31.

The following proposition is frequently used.

*110·32.

*110·33.

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 have been more difficulty in dealing with types, and because the existence of (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.

Dem.

*110·34.

*110·35.

*110·351.

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 unless both and are existent cardinals. It is chiefly useful in avoiding the necessity of the hypothesis , in such propositions as the commutative and associative laws.

*110·4.

The following propositions, down to *110·411 inclusive, are concerned with types. They are not referred to in the sequel.

*110·401.

Dem.

*110·402.

Dem.

*110·403.

*110·404.

*110·41.

Dem.

*110·411.

It will be observed that the following proposition (*110·42) requires no hypothesis. This is owing to *110·4 and *102·74.

*110·42.

Dem.

*110·43.

*110·44.

Dem.

The above proposition depends upon the fact that is typically ambiguous, even when and 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.

Dem.

*110·501.

*110·51.

It is not necessary to the truth of the above proposition that and should be cardinals. If either is not a cardinal, and are both .

The following propositions lead to the associative law (*110·56).

*110·52.

Dem.

*110·521.

*110·53.

*110·531.

*110·54.

Dem.

*110·541.

Dem.

*110·55.

*110·551.

*110·56.

Dem.

This is the associative law for arithmetical addition. It will be seen that, like the commutative law, it does not require that , , should be cardinals.

*110·561.

*110·57.

The following propositions, concerning the addition of 0 or 1, are used frequently in dealing with inductive cardinals (*120).

*110·6.

Dem.

When is a typically definite cardinal, is the same cardinal rendered typically ambiguous; when is a typically ambiguous cardinal, , is . In place of the above proposition, we might write ; this would be true whenever the ambiguity of was so determined as to make it significant. But the above form gives more information.

*110·61.

Dem.

In this proposition, is typically ambiguous; hence we escape the necessity of putting on the right, as we should have to do if were typically definite. We can deduce *110·61 from *110·6 as follows:

We have to travel via in this proof, in order to avoid the possibility of a typical determination of which would make . It is for the same reason that we cannot put ""; for if the first is determined to a type in which , while the second is not, this equation becomes false.

*110·62.

Dem.

*110·63.

Dem.

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.

Dem.

The proposition 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 is a type, and . Then if is a finite class, . Hence . Hence in all types. But will exist in all types higher than that of . If on the other hand the number of entities in is infinite, we shall have Hence in this case the above proposition will be true universally.

*110·632.

Dem.

*110·64.

*110·641.

*110·642.

*110·643.

Dem.

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.

Dem.

*110·71.

Dem.

The above proof depends upon the fact that "" and "" 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 , i.e. for . This is why the use of *100·3 is legitimate.

*110·72.

Dem.

*111. DOUBLE SIMILARITY.

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 is a class of classes, the number of members of is an arithmetical property of , but it is obvious that this is not determined by the number of members of , but requires also a knowledge of the numbers of members of members of . For example, let consist of the two members and , and let consist of and . Then ; but in order to be able to infer , we require , and or 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 and , insures that all the arithmetical properties of and are the same, e.g. we have (in particular) and . This relation we denote by "," which is to be read as one symbol. It is defined as follows: We define first the class of "double correlators" of and , which we denote by "," and of which the definition is

*111·01.

so that

We then define "" as meaning that is not null, i.e. that there is at least one double correlator of and .

To illustrate the nature of a double correlator, let us suppose that consists of the two classes and , and that consists of , , while consists of , , . Similarly let consist of and while consists of , and consists of , , . Now let correlate each with the having the same two suffixes. Then is a one-one, and its converse domain is . Moreover (which is ) = , and , so that . Thus is a double correlator according to the definition.

The essential characteristic of a double correlator is that (1) is a correlator of and , (2) is a correlator of and . If we write in place of , then if , we have ; moreover is a correlator of and . Thus and are similar classes of similar classes. They are not merely this, however, for we not only know that is similar to , but we know a particular correlator of and , namely . This is essential to the use of double similarity, as will appear shortly.

Let us consider the relation between and which consists in their being similar classes of similar classes. This means that there is a correlator of and , such that, if , is similar to . That is to say, we are to consider the hypothesis or, as it may be more briefly expressed, Let us assume . If we attempt to prove (say) that is similar to , we find that we are forced to assume the multiplicative axiom, unless and are finite. This necessity arises as follows. Let us put where "" stands for "correspondence." Then we know that whenever , is not null. Further it is easy to prove that, if and are classes of mutually exclusive classes, and if we can pick out one representative member of for each value of which is a member of , then the relational sum of all these representative correlations gives us a correlator of and . That is, we have

But in order to infer hence , we need , i.e. we need to be able to pick out a particular correlator for each pair of similar classes and . 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 , 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 is given as of the form , where . If the multiplicative axiom is assumed, but in general not otherwise, we have (*111·5)

In the present number, we shall begin with various properties of double correlators. We prove (*111·11) that is a double correlator of and when, and only when, is a correlator of and , and is a correlator of and . We prove (*111·112) that in the same hypothesis, . We prove (*111·13) that is a double correlator of with itself; that (*111·131) if is a double correlator of and , is a double correlator of and ; that (*111·132) if , are double correlators of with and of with respectively, is a double correlator of with . Hence it follows (*111·45·451·452) that double similarity is reflexive, symmetrical, and transitive.

We then proceed (*111·2—·34) to consider , where it is to be supposed that is a correlator of and , and that is similar to if . We prove

*111·32.

Thus in the case supposed, is a double correlator of and . Thus

*111·322.

We then proceed (*111·4—·47) to various propositions on "," and finally (*111·5·51·53) state three propositions which assume the multiplicative axiom, namely

*111·5. If , , then .

*111·51. In the same case, , i.e. if and are similar classes of mutually exclusive similar classes, their sums are similar.

*111·53. In the same case, if , , . Hence the multiplicative axiom implies that two classes of mutually exclusive classes each of which has terms, have the same number of terms in their sum.

*111·01.

*111·02.

*111·03.

*111·1.

*111·11.

Dem.

*111·111.

Dem.

*111·112.

The two following propositions are useful lemmas for the case when is replaced (as it often is) by .

*111·12.

Dem.

*111·121.

Dem.

*111·13.

Dem.

*111·131.

Dem.

*111·132.

Dem.

*111·14.

Dem.

*111·15.

Dem.

*111·16.

Dem.

*111·18.

Dem.

The class is important, being the class of Cantor's "Belegungen," used by him to define exponentiation; we have in fact Thus the above proposition shows that is less than or equal to ; and since, whenever it is not zero, , it is less than or equal to

The following propositions lead up to *111·32·33·34:

*111·2.

*111·201.

*111·202.

*111·21.

*111·211.

*111·22.

Dem.

*111·221.

Dem.

*111·23.

Dem.

*111·24.

Dem.

*111·25.

*111·3.

Dem.

*111·31.

Dem.

*111·311.

*111·313.

Dem.

*111·32.

Dem.

*111·321.

*111·322.

*111·33.

Dem.

*111·34.

Dem.

The following propositions are concerned with the elementary properties of "." It will be seen that they are closely analogous to those of "."

*111·4.

*111·401.

Dem.

*111·402.

*111·43.

*111·44.

*111·45.

*111·451.

*111·452.

*111·46.

*111·47.

Dem.

*111·5.

*111·51.

*111·52.

Dem.

*111·53.

*112. THE ARITHMETICAL SUM OF A CLASS OF CLASSES.

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 is a class of mutually exclusive classes, the number of will be the sum of the numbers of members of ; i.e. if we write "" for the sum of the numbers of members of , But when the members of are not mutually exclusive, a term which is a member of two members (say and ) of has to be counted twice over in obtaining the arithmetical sum of , whereas in the logical sum is only counted once. Thus we need a construction which shall duplicate , taking it first as a member of , and then as a member of . This is effected if we replace first by , and then by . In fact, has the kind of arithmetical properties which we mean to secure when we speak of " considered as a member of "—a phrase which, as it stands, does not serve our purpose, for is simply however we may choose to consider it. Thus we replace by and by and so on; i.e. (using *85·5), we replace by and by and so on. These new classes are similar to and 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 . Thus we put

With regard to the second of these definitions, it is to be observed that is not a function of , unless no two members of are similar; for cannot contain the same number twice over. For the same reason, if is a class of cardinals, and we define "," we do not get what 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 "" as follows: Take a class of classes , consisting of one class having each number which is a member of , i.e. let be a selection from ; then will have the required number of terms. i.e. we might put 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.

This is an extension of *110·32.

*112·17.

The chief point in the above proposition is that it does not require , .

*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.

whence we derive the proposition

*112·24.

I.e. assuming the multiplicative axiom, two classes which each consist of classes of terms each have the same number of terms in their sum. This number would naturally be defined as multiplied by , 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 mutually exclusive sets of terms, cannot be proved in general without the multiplicative axiom.

The remaining propositions of this number give properties of in special cases. We prove that (*112·3), that (*112·321), that (*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.

*112·43.

*112·01.

*112·02.

*112·1.

*112·101.

*112·102.

Dem.

*112·103.

*112·11.

*112·12.

*112·13.

*112·14.

Dem.

*112·15.

*112·151.

Dem.

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.

Dem.

In the following proposition, we have a double correlator of a sort which will frequently occur in cardinal arithmetic, namely with its converse domain limited, where 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 is a correlator whose converse domain includes and has as a member, . Thus 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.

Dem.

*112·16.

*112·17.

Dem.

*112·18.

Dem.

*112·2.

Dem.

*112·21.

*112·22.

*112·23.

Dem.

*112·231.

Dem.

*112·24.

Dem.

*112·3.

*112·301.

Dem.

*112·302.

Dem.

Thus if is a member of a class of classes, it does not affect the value of their arithmetical sum.

*112·303.

Dem.

*112·304.

Dem.

*112·31.

Dem.

*112·311.

Dem.

*112·32.

Dem.

*112·321.

*112·33.

*112·331.

*112·34.

Dem.

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 in the above proposition.

*112·341.

Dem.

*112·35.

Dem.

Similar propositions can obviously be proved for any finite number of summands.

*112·4.

Dem.

*112·41.

Dem.

*112·42.

Dem.

*112·43.

Dem.

The above is the associative law for arithmetical addition.

*113. ON THE ARITHMETICAL PRODUCT OF TWO CLASSES OR OF TWO CARDINALS. — FOOTNOTES:

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 and , and thence the product of two cardinals and as the number of terms in the product of and when has terms and has 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 and , which we denote by [5], is the class of all ordinal couples which take their referent from and their relatum from , i.e. it is the class of all such relations as , where and . For a given , the class of couples we obtain is , which is similar to ; and the number of such classes, for varying , is . Thus we have classes of couples, and is the logical sum of these classes of couples. The class of such classes as , where , is important again in connection with exponentiation; we have , whence the class of such classes, when is varied among the 's, is , and which we take as the definition of .

We represent the arithmetical product of and by . This, as well as , is defined in terms of exactly as, in *110, the sum was defined in terms of .

The present number contains many propositions which belong to the theory of rather than (specially) of ; 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 or on its sum, which is . Thus in the present number, the early propositions, down to *113·118, deal mainly with logical properties of and ; the following propositions, down to *113·13, deal mainly with arithmetical properties of ; the propositions *113·14—·191 are concerned mainly with arithmetical properties of ; *113·2—·27 deal with the simpler properties of ; *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.

This merely embodies the definition of .

*113·105.

This proposition is especially useful in dealing with exponentiation (*116).

*113·114.

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.

This proposition is chiefly useful in the analogous theory of ordinal products (*165, *166), where it enables us to apply *74·773. Unless , we have , and unless (*113·116).

*113·12.

I.e. unless is null, consists of mutually exclusive classes each having members.

*113·127.

This is an important proposition, since it gives a double correlator of with whenever simple correlators of with and of with are given. It leads at once to

*113·13.

This proposition is fundamental in the theory of multiplication, since it shows that the number of members of depends only upon the numbers of members of and . It is also fundamental in the theory of exponentiation, as will appear in *116.

*113·141.

This is the source of the commutative law of multiplication (*113·27).

*113·146.

This connects our present theory of multiplication with the theory of selections.

We come next to propositions concerning . We have

*113·204.

The use of this proposition, like that of *110·4, is for avoiding trivial exceptions.

*113·23.

*113·25.

This proposition enables us to infer propositions on products of cardinals from propositions on products of classes, and is therefore constantly used.

*113·27.

This is the commutative law of cardinal multiplication.

The chief proposition using the multiplicative axiom is

*113·31.

I.e. assuming the multiplicative axiom, the sum of the numbers of members in classes of terms is . If we had taken this sum as defining , almost all propositions on multiplication would have required the multiplicative axiom. The advantage of is that, given and , we can construct a double correlator of with , 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.

whence, using also the commutative law, we easily deduce

*113·43.

But the distributive law also holds when, instead of enumerated summands , or , , the summands are given as the members of a class , which may be infinite. We have

*113·48.

whence, using the definitions of *112, we find

*113·491.

This is an extension of the distributive law to the case where the number of summands may be infinite.

The associative law

*113·54.

is proved without any difficulty.

We prove next that when, and only when, or , , being existent cardinals (*113·602); that a cardinal is unchanged when it is multiplied by 1 (*113·62·621); that (*113·66) and that (*113·671).

*113·02.

*113·03.

*113·04.

*113·05.

In relation to types, *113·03·04·05 call for similar remarks to those made in *110 for addition.

*113·1.

*113·101.

*113·102.

Dem.

*113·103.

*113·104.

*113·105.

Dem.

*113·106.

*113·107.

*113·11.

Dem.

*113·111.

*113·112.

Dem.

*113·113.

*113·114.

*113·115.

Dem.

*113·116.

*113·117.

*113·118.

*113·12.

*113·121.

*113·122.

*113·123.

*113·124.

Dem.

*113·125.

*113·126.

Dem.

*113·127.

*113·128.

*113·13.

*113·14.

*113·141.

*113·142.

Dem.

*113·143.

Dem.

*113·144.

Dem.

Note to *113·144. In virtue of *113·143 and *55·61 we have

At a later stage (in *150) we shall put

Thus we shall have, anticipating this notation,

Hence we have

*113·145.

*113·146.

*113·147.

Dem.

The advantage of this proposition is that it exhibits the correlator of and as a function of .

*113·148.

Dem.

*113·15.

Dem.

*113·151.

*113·152.

Dem.

The following proposition is only significant when and are classes of relations. It is used in relation-arithmetic (*172·34).

*113·153.

Dem.

*113·16.

Dem.

*113·17.

Dem.

*113·171.

Dem.

Note that the hypothesis is only significant when and are of the same type.

*113·172.

Dem.

*113·18.

Dem.

*113·181.

Dem.

*113·182.

*113·183.

Dem.

*113·19.

Dem.

*113·191.

Dem.

*113·2.

*113·201.

*113·202.

Dem.

*113·203.

*113·204.

*113·205.

*113·21.

*113·22.

Dem.

*113·221.

*113·222.

Dem.

*113·23.

Dem.

*113·24.

*113·25.

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.

*113·26.

Dem.