*83. SELECTIONS FROM CLASSES OF CLASSES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
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.