Volume 1 / Part 3 — *111. DOUBLE SIMILARITY.
Reading edition
*111. DOUBLE SIMILARITY.
Summary of *111.
The arithmetical properties of a class, so far as these do not require or assume that it is a class of classes, are the same for any similar class. But a class of classes has many arithmetical properties which it does not share with all similar classes of classes. For example, if is a class of classes, the number of members of is an arithmetical property of , but it is obvious that this is not determined by the number of members of , but requires also a knowledge of the numbers of members of members of . For example, let consist of the two members and , and let consist of and . Then ; but in order to be able to infer , we require , and or or some such further datum. The relation of "double similarity," to be defined in the present number, is a relation between classes of classes, which, when it holds between and , insures that all the arithmetical properties of and are the same, e.g. we have (in particular) and . This relation we denote by "," which is to be read as one symbol. It is defined as follows: We define first the class of "double correlators" of and , which we denote by "," and of which the definition is
*111·01.
so that
We then define "" as meaning that is not null, i.e. that there is at least one double correlator of and .
To illustrate the nature of a double correlator, let us suppose that consists of the two classes and , and that consists of , , while consists of , , . Similarly let consist of and while consists of , and consists of , , . Now let correlate each with the having the same two suffixes. Then is a one-one, and its converse domain is . Moreover (which is ) = , and , so that . Thus is a double correlator according to the definition.
The essential characteristic of a double correlator is that (1) is a correlator of and , (2) is a correlator of and . If we write in place of , then if , we have ; moreover is a correlator of and . Thus and are similar classes of similar classes. They are not merely this, however, for we not only know that is similar to , but we know a particular correlator of and , namely . This is essential to the use of double similarity, as will appear shortly.
Let us consider the relation between and which consists in their being similar classes of similar classes. This means that there is a correlator of and , such that, if , is similar to . That is to say, we are to consider the hypothesis or, as it may be more briefly expressed, Let us assume . If we attempt to prove (say) that is similar to , we find that we are forced to assume the multiplicative axiom, unless and are finite. This necessity arises as follows. Let us put where "" stands for "correspondence." Then we know that whenever , is not null. Further it is easy to prove that, if and are classes of mutually exclusive classes, and if we can pick out one representative member of for each value of which is a member of , then the relational sum of all these representative correlations gives us a correlator of and . That is, we have
But in order to infer hence , we need , i.e. we need to be able to pick out a particular correlator for each pair of similar classes and . This, however, cannot be done in general without assuming the multiplicative axiom. It follows that we must not define two classes as having double similarity when , but must give a definition which enables us to specify a particular correlator for each pair of similar classes. This is what is effected by the above definition of double correlators, where our is given as of the form , where . If the multiplicative axiom is assumed, but in general not otherwise, we have (*111·5)
In the present number, we shall begin with various properties of double correlators. We prove (*111·11) that is a double correlator of and when, and only when, is a correlator of and , and is a correlator of and . We prove (*111·112) that in the same hypothesis, . We prove (*111·13) that is a double correlator of with itself; that (*111·131) if is a double correlator of and , is a double correlator of and ; that (*111·132) if , are double correlators of with and of with respectively, is a double correlator of with . Hence it follows (*111·45·451·452) that double similarity is reflexive, symmetrical, and transitive.
We then proceed (*111·2—·34) to consider , where it is to be supposed that is a correlator of and , and that is similar to if . We prove
*111·32.
Thus in the case supposed, is a double correlator of and . Thus
*111·322.
We then proceed (*111·4—·47) to various propositions on "," and finally (*111·5·51·53) state three propositions which assume the multiplicative axiom, namely
*111·5. If , , then .
*111·51. In the same case, , i.e. if and are similar classes of mutually exclusive similar classes, their sums are similar.
*111·53. In the same case, if , , . Hence the multiplicative axiom implies that two classes of mutually exclusive classes each of which has terms, have the same number of terms in their sum.
*111·01.
*111·02.
*111·03.
*111·1.
*111·11.
Dem.
*111·111.
Dem.
*111·112.
The two following propositions are useful lemmas for the case when is replaced (as it often is) by .
*111·12.
Dem.
*111·121.
Dem.
*111·13.
Dem.
*111·131.
Dem.
*111·132.
Dem.
*111·14.
Dem.
*111·15.
Dem.
*111·16.
Dem.
*111·18.
Dem.
The class is important, being the class of Cantor's "Belegungen," used by him to define exponentiation; we have in fact Thus the above proposition shows that is less than or equal to ; and since, whenever it is not zero, , it is less than or equal to
The following propositions lead up to *111·32·33·34:
*111·2.
*111·201.
*111·202.
*111·21.
*111·211.
*111·22.
Dem.
*111·221.
Dem.
*111·23.
Dem.
*111·24.
Dem.
*111·25.
*111·3.
Dem.
*111·31.
Dem.
*111·311.
*111·313.
Dem.
*111·32.
Dem.
*111·321.
*111·322.
*111·33.
Dem.
*111·34.
Dem.
The following propositions are concerned with the elementary properties of "." It will be seen that they are closely analogous to those of "."
*111·4.
*111·401.
Dem.
*111·402.
*111·43.
*111·44.
*111·45.
*111·451.
*111·452.
*111·46.
*111·47.
Dem.
*111·5.
*111·51.
*111·52.
Dem.
*111·53.