Principia mathematica, vol. 1 (of 3)

The subject to be considered in this section is important chiefly in connection with multiplication, both cardinal and ordinal. In order to get a definition of multiplication which is not confined to the case where the number of factors is finite, we have to seek a construction by which, from a given class of classes, say, we construct another class which, when is finite, has that number of terms which, in the usual elementary sense, is the product of the numbers of terms in the various classes which are members of , and which, whether is finite or not, obeys as many as possible of the formal laws of multiplication. The usual elementary sense of multiplication is derived from addition; that is to say, is to be the number of terms in , where is a class of mutually exclusive classes each having members, or vice versa. This sense can be extended to any finite number of factors, but not to an infinite number of factors; hence for a number of factors which may be infinite we require a different definition, and this is derived from the theory of selections.

Selections are of two kinds, selections from classes of classes, and selections from relations. The latter is the more general notion, from which the former is derived. But as the former is an easier notion, we will begin by explaining selections from classes of classes.

Given a class of classes , a class is called a selected class of when is formed by choosing one term out of each member of . For example, if consists of two members, and , and if and , then is a selected class of . If every constituency elects a local man, Parliament is a selected class of the constituencies. If is a class of mutually exclusive classes, i.e. a class no two of whose members have any member in common, then a selected class consists of only one term from each member of ; i.e. is a selected class if But if is not a class of mutually exclusive classes, this does not hold necessarily; for a term which is a member of both and (where ) may be chosen as the representative of , while some other term may be chosen as the representative of , so that two members of may belong to the selected class. Again, if is a class of mutually exclusive classes, the relation of the representative to its class must be one-one, because, since no term belongs to two classes which are members of , no term can be the representative of two classes. But when is not a class of mutually exclusive classes, a term which belongs to two classes and may be chosen as the representative of both. Thus the relation of the representative to its class may be only one-many, not one-one.

The relation of the representative to its class may be called a selective relation. A selective relation of is one which selects, from every class which is a member of , a certain member as the representative of ; that is, we have, if is the selective relation, This condition is equivalent to

It will be seen that, if , may be any member of , and we get a different for each different member of . Thus if we keep the representatives of all the other members of unchanged, the number of selective relations to be obtained by varying the representative of is the number of members of . Hence the number of selective relations altogether may be fitly defined as the product of the numbers of terms possessed by the various members of . In case is finite, this agrees with the usual definition of multiplication; and whether is finite or infinite, the product so defined obeys all the formal laws of multiplication.

To illustrate the notion of selective relations, let us take a very simple case, the case where consists of two classes and , each of which has two members. Let and be the members of , and the members of . We assume , , . Then the selective relations of are the following: Thus they are four in number, i.e. the number of members of is the product of the number of members of and the number of members of . A similar process would show that our definition of the product agrees with the usual definition in any case in which all the numbers concerned are finite.

Selections from relations are an obvious generalization of selections from classes of classes. We had above We put, generally, which we derive from the definition This is the fundamental definition in the subject of selections. We have, in virtue of this definition, When , we may call the class of selections from . Thus generally, is the class of selections from provided ; and if this condition is not fulfilled, . We may call the class the class of "-selections from ." The class of "-selections from " will be what we previously called the class of "selective relations of ." It will be observed that we have Thus if is a class of mutually exclusive classes, selects one from each of these classes, and is therefore a selective class of ; hence in this case

In Cardinal Arithmetic, is the important notion, and the more general notion is seldom required. In Ordinal Arithmetic, is the important notion. It will be seen that Thus is only significant when is a class of relations; in this case we have Thus chooses a representative member of the field of every member of . The most important case is when is of the form , where is a serial relation whose field consists of serial relations. Then becomes the field of a relation which may be defined as the ordinal product of the relations composing ; in this way we get an infinite ordinal product analogous to the infinite cardinal product. This will be explained at a later stage (*172).

, as above defined, is the class of one-many relations contained in and having for their converse domain. We know of no proof that there always are such relations when . In fact, the proposition is equivalent to the "multiplicative axiom," i.e. to the axiom that, given any class of mutually exclusive classes, none of which is null, there is at least one class formed of one member from each of these classes. (This equivalence is proved in *88·36, below.) It is also equivalent to Zermelo's axiom[61], which is hence also it is equivalent to the proposition that every class can be well-ordered. In the absence of evidence as to the truth or falsehood of these various propositions, we shall not assume their truth, but shall explicitly introduce them as hypotheses wherever they are relevant.

In the present section, we shall begin (*80) by considering such properties of as do not depend upon any hypothesis as to . We shall then (*81) proceed to consider such further properties of as result from the hypothesis . This hypothesis is important, because it is verified in many of the applications we wish to make, and because it leads to important properties of which are not true in general when is not subject to any hypothesis. These special properties are mostly due to the fact that when is a many-one relation, consists of one-one relations (not merely of one-many relations, as it does in the general case). This is proved in *81·1. We then (*82) proceed to consider the case of relative products, i.e. (. It will appear that, with a suitable hypothesis, ( and . In the following number (*83) we apply the results of *80 to the particular case where is replaced by , which is the important case for cardinal arithmetic. In *84 we apply the propositions of *81 to the case where is replaced by , and where, therefore, we have the hypothesis . This hypothesis is equivalent to the hypothesis that no two members of have any members in common, i.e. that When fulfils this hypothesis, it is a class of mutually exclusive classes. For classes of mutually exclusive classes we adopt the notation "." It is shown in *84·14 that a is one for which we have . When is a , is a one-one relation, and .

Also in this case consists of all classes formed of one member from each member of , i.e. all classes such that In *85, we prove various important propositions, of which the chief is a form of the associative law[62], namely Finally, in *88, we consider the question of the existence of selections. This cannot in general be proved when is an infinite class. The assumption that is never null unless one member of is null is equivalent to various other assumptions, for example to the assumption that every class can be well-ordered. One of these equivalent assumptions is called the "multiplicative axiom." This axiom is equivalent to the assumption that an arithmetical product cannot be zero unless one of its factors is zero, and is regarded by some mathematicians as a self-evident truth. This can be proved when the number of factors is finite, i.e. when is a finite class, but not when the number of factors is infinite. We have not assumed its truth in the general case where it cannot be proved, but have included it in the hypotheses of all propositions which depend upon it.