Volume 1 / Part 2 — *85. MISCELLANEOUS PROPOSITIONS.
Reading edition
*85. MISCELLANEOUS PROPOSITIONS.
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 , .
The above proposition is used in connection with "families" (*97·31).
*85·14.
Dem.
*85·21·22 are lemmas for *85·24, which, with *85·26, is required for *85·27.
*85·21.
*85·22.
Here can also be written . The brackets are omitted because no other meaning is possible.
Dem.
*85·24.
Dem.
The following propositions are lemmas for *85·26.
*85·241.
Dem.
*85·243.
Dem.
*85·244.
Dem.
*85·245.
Dem.
*85·25.
*85·26.
Dem.
*85·27.
*85·28.
The following proposition is a lemma for *85·31.
*85·3.
The conditions of significance here and in *85·31·32·33·34 require .
Dem.
The following propositions, down to *85·42 inclusive, deal with circumstances under which we can infer from . *85·32·33·34 are not subsequently used; the remainder are used in proving *85·43.
*85·31.
Dem.
*85·32.
Dem.
*85·33.
The proof proceeds exactly as in *85·32.
*85·34.
The following propositions, *85·4·41·42, are lemmas for *85·43·44, which latter are of fundamental importance, since they are the source of the associative law in cardinal arithmetic.
*85·4.
*85·41.
Dem.
*85·42.
Dem.
*85·43.
Dem.
*85·44.
The following proposition is used in connection with cardinal multiplication (*114·301).
*85·45.
Dem.
The purpose of the following propositions, down to *85·55, is to show how to get from a class of classes a class of selections having the same number of terms as . For this purpose we introduce a new notation, representing a rather important analysis of the couples contained in a given relation. A couple is contained in a relation when ; thus if, keeping fixed, we form the class of couples , all these couples are contained in . We put
*85·5.
Then . Also is the class of all couples contained in , and . We shall now prove that , so that every member of can be derived from a member of , and the problem of the existence of is reduced to that of the existence of selections from a class of mutually exclusive existent classes.
*85·51.
*85·52.
*85·53.
Dem.
*85·54.
Dem.
The following proposition is frequently useful.
*85·55.
Dem.
*85·56.
*85·6.
Dem.
The following proposition is frequently employed.
*85·601.
Dem.
*85·61.
*85·62.
*85·63.
Dem.
Note. ( is "Zermelo's axiom." The above proposition shows that this is true if which again is true if in virtue of *84·412. The last of these is the "multiplicative axiom," which is thus shown to imply "Zermelo's axiom."
The following propositions lead up to *85·72, which is used in the theory of double similarity (*111·3).
*85·7.
Dem.
*85·701.
*85·702.
*85·71.
This proposition asserts that if we can select one sub-class out of each member of (where is a class of classes), then selections from the sub-classes so obtained are selections from .
*85·72.
Dem.
The following proposition is a lemma employed in the theory of double similarity (*111·313).
*85·81.
Dem.