Volume 1 / Part 2 — *83. SELECTIONS FROM CLASSES OF CLASSES.

*83. SELECTIONS FROM CLASSES OF CLASSES.

Summary of *83.

In this number, the general propositions which have been proved for are to be applied to the important special case where is . In this case, we have selections from classes of classes: if , picks out a representative from each class which is a member of ; i.e. we have

The propositions of this number result from those of previous numbers either immediately, by the substitution of for , or by the use of propositions of *62, notably (*62·2), and (*62·3).

The propositions of the present number follow, in the main, the same course as those of *80, with substituted for (except that the special forms of propositions before *80·2 are not given). We have first a set of propositions resulting immediately from early propositions of *80. Of these the most used are:

*83·11.

This leads to the proposition that an arithmetical product is null if one of its factors is null. (We cannot prove the converse universally without assuming the multiplicative axiom.)

*83·15.

Thus is a unit class. This is the source of the proposition , where is a cardinal (cf. note to *83·15).

*83·2.

Here is the "representative" of .

*83·21.

We have next a set of propositions (*83·4—·44) on selections from unit classes and classes of unit classes. We have

*83·41.

This leads to the proposition that a product of one factor is equal to that factor.

*83·43.

This leads to

*83·44.

whence it follows that a product of factors, each of which is one, is one. This holds even if the number of factors is infinite or zero.

We have next a set of propositions (*83·5—*·58) on changing the representative of a class, and on selections from a class of classes some of which are unit classes. These propositions are seldom referred to in the sequel.

We have next (*83·6—·74) a set of propositions on the domains of selections, i.e. on the class . We have

*83·66.

(The hypothesis here cannot be dispensed with unless we assume the multiplicative axiom.)

*83·7.

*83·71.

We have next two propositions (*83·8·81) on the types of and . The type of is the same as that of (*83·81).

The last set of propositions in this number (*83·9—·904) deals with the existence of selections. We have

*83·9.

*83·901.

*83·904.

From these propositions we shall deduce by mathematical induction that whenever is a finite class, exists unless (cf. *120·62). Thus a product consisting of a finite number of factors (which may themselves be either finite or infinite) can only vanish if one of the factors vanishes.

*83·1.

Dem.

*83·11.

*83·12.

*83·13.

*83·14.

*83·15.

In virtue of this proposition, the product of 0 cardinal numbers is 1—a proposition of which a particular case, namely , is familiar. This arithmetical proposition results from the above as follows. We shall define the product of the numbers of members of as the number of members of . Thus when , the number of members of is a product of 0 factors. Now by the above proposition, has one member, namely . Hence a product of 0 factors is 1.

*83·16.

*83·2.

*83·21.

*83·22.

*83·23.

*83·24.

*83·25.

*83·26.

*83·27.

*83·271.

*83·28.

*83·29.

*83·3.

*83·31.

*83·4.

*83·41.

This proposition shows that a cardinal product of one factor is equal to that one factor. For the number of members of is the product of the numbers of members of members of , i.e. it is a product whose only factor is the number of members of . By the above proposition, this product is equal to the number of members of .

*83·42.

Dem.

This proposition shows that a cardinal product whose factors are all 1 is 1. For is a class whose members are all unit classes, and thus the number of members of is the product of a number of 1's; and by the above proposition, is a unit class, its sole member being . This result is rendered more explicit by *83·43·44.

*83·43.

Dem.

*83·44.

*83·5.

Dem.

It follows from this proposition that if is a class of classes for which there are selections, and if one member (not null) be added to , there are still selections from the resulting class of classes.

*83·51.

*83·52.

*83·54.

Dem.

*83·55.

Dem.

*83·56.

Dem.

The following proposition is used in the theory of cardinal multiplication (*114·41).

*83·57.

Dem.

*83·58.

Dem.

This proposition shows that in a product any number of factors each equal to 1 may be omitted without altering the value of the product.

The following propositions, down to *83·74, are concerned with the domains of selective relations, i.e. with the selected classes.

*83·6.

Dem.

*83·61.

Dem.

*83·62.

*83·63.

Dem.

*83·64.

Observe that the hypothesis required here is , not as in *83·63.

Dem.

The following proposition is used in connection with cardinal multiplication (*115·14).

*83·641.

Dem.

*83·65.

Dem.

*83·66.

Dem.

*83·7.

*83·71.

Dem.

*83·72.

Dem.

*83·73·731 are lemmas for *83·74.

*83·73.

Dem.

*83·731.

Dem.

*83·74.

Dem.

*83·8.

Dem.

*83·81.

Dem.

*83·9.

*83·901.

*83·902.

*83·903.

*83·904.

*83·9·904 leads to an inductive proof (to be given later) of whenever is a finite class of classes none of which is .