Volume 1 / Part 3 — SECTION A. DEFINITION AND LOGICAL PROPERTIES OF CARDINAL NUMBERS.
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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 .