*85. MISCELLANEOUS PROPOSITIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *85.
In this number certain important propositions are proved, and the other propositions of this number are mainly lemmas. The most important propositions are the following:
*85·1 and *85·14, which show that if is a , then the domains of are the same as the domains of , and is similar to , thus reducing the problem of selections from many-one relations to that of selections from classes of classes.
*85·27 and *85·43, which show that if , consists of the relational sums of the domains of and is similar to ; i.e. the class of -selections from is similar to the class obtained as follows: take the members of one by one, and form the -selections of each; we thus obtain a class of classes, each class being of the form , where ; we then make a selection from this class of classes; this selection is a member of ; the number of such selections is the same as the number of .
*85·28 and *85·44, which are special cases of *85·27 and *85·43, but more useful than these. *85·44 is the source of the associative law in cardinal multiplication; it states that, if is a , has the same number of members as . (On associative laws in general, see the notes to *42·1·11.) That is to say, if we form the class of selective relations ( for every which is a member of , and then form the class of selective relations for , we get the same number of terms as if we proceeded to form the class of selective relations for . The way in which this proposition yields the associative law of multiplication may be explained as follows. We shall define the product of the numbers of members of as the number of . Thus e.g. if the numbers of the members of are , the number of is . Suppose the other members of are and , and that and again have three members each. Then the number of is the product of the numbers of , , , i.e. it is the product of , and .
But the numbers of the members of are
Thus the number of is
Hence *85·44 enables us to conclude that which is a case of the associative law. In fact *85·44 gives us this law in its general form, when the number of brackets, and of factors in each bracket, may be infinite or finite indifferently.
Another important pair of propositions is *85·53·54. These enable us to reduce the problem of selections for any relation to the problem of selections from a class of classes. The method is as follows: Given any term , form the class of ordered couples of which is relatum while the referent is a term having the relation to . Call this class of couples . Form this class for every which is a member of ; we thus obtain a class of classes, namely . Then the number of selections from this class of classes is the same as the number of .
We have one other important pair of propositions in this number, namely *85·61·63. These show that what is called "Zermelo's axiom" is equivalent to what is called the "multiplicative axiom." Zermelo's axiom[63] is to the effect that if is any class, is never null, i.e. . The "multiplicative axiom" is to the effect that if , there is at least one class formed by taking one representative from each member of , which is equivalent to
In *85·63, these two axioms are shown to be equivalent. From Zermelo's theorem[64] it follows that both are equivalent to the assumption that every class can be well-ordered. This will be proved later (*258).
The above-mentioned propositions, stated symbolically, are as follows:
*85·1.
*85·14.
*85·27.
*85·28.
*85·43.
*85·44.
The following propositions depend upon the definition
*85·5.
I.e. is the class of all couples whose relatum is y while the referent has the relation to . We then have
*85·53.
giving a construction for by means of , and
*85·54.
which reduces the question of the existence of -selections to that of the existence of -selections.
*85·61.
This proposition gives a construction for any -selection in terms of an -selection from a , and reduces the question of the existence of the former to that of the existence of the latter. A particularly important case is when . This is considered in
*85·63.
*85·1.
Dem.
*85·11.
Dem.
*85·111.
*85·112.
*85·12.
Dem.
This proposition is used in connection with ordinal multiplication (*173·14).
*85·13.
Dem.
In the above proposition, the hypothesis required as to by *82·231 is only ; but since , .