Volume 1 / Part 2 — *73. SIMILARITY OF CLASSES.
Reading edition
*73. SIMILARITY OF CLASSES.
Summary of *73.
Two classes and are said to be similar when there is a one-one relation whose domain is and whose converse domain is . We express " is similar to " by the notation "." When two classes are similar, they have the same cardinal number of terms: it is this fact which gives importance to the relation of similarity.
We have
The relation of similarity is that of the domain of a to the converse domain, i.e. it is the relative product of and , or, what comes to the same thing, it is the relative product of and .
Most of the properties of similarity result immediately from those of one-one relations and offer no difficulty of any kind.
When there are relations which correlate 's with 's so as to make similar to , we denote the class of such relations by "." Thus we have
When, as in this case, we have a descriptive double function closely connected with a relation, we shall make it a practice to distinguish the descriptive double function by a bar.
It is to be observed that "," like and and and , is ambiguous as to type, and only acquires a definite meaning when the types of its domain and converse domain are specified. The domain and the converse domain may or may not be of the same type, i.e. "" may or may not be a homogeneous relation. This enables us to speak of two classes of different types as having the same number of terms. We shall return to this point in connection with cardinal numbers (cf. especially *102—*106).
The propositions of the present number are important, and are very frequently referred to throughout cardinal arithmetic. In order to prove that two classes and have the same cardinal number of terms, it is generally necessary, in the fundamental arithmetical propositions with which we are concerned, actually to construct a relation such that . Such relation will be called a correlator of and . It will usually be obtained by taking some relation for which we have (, and limiting the converse domain to , so that is the required correlator. Very frequently we shall have , not , but will be such that .
Among the more important propositions of the present number are the following:
*73·142.
I.e. is a correlator of and if (1) is one-one, (2) is contained in the converse domain of , (3) is the class of those terms which have the relation to members of .
*73·2.
This results immediately from the definition.
*73·22.
*73·3.
*73·31.
*73·32.
The above three propositions show that similarity is reflexive, symmetrical and transitive.
*73·36.
*73·41.
Thus every class is similar to a class of higher type, and consisting wholly of unit classes.
*73·45.
Thus 1 is the class of all classes similar to any unit class.
*73·48.
Thus 0 is the class of all classes similar to the null-class.
*73·611.
This proposition is very often useful. For arithmetical purposes, we often wish to obtain mutually exclusive classes. Now whether or not and be mutually exclusive, and are mutually exclusive provided . Thus by means of the above proposition we can always construct mutually exclusive classes each similar to a given class, i.e. each having some assigned number of members.
*73·71.
This proposition is fundamental in the theory of addition.
*73·88.
I.e. "if is similar to a part of , and is similar to a part of , then is similar to ." This is the Schröder-Bernstein theorem. The proof given below is due to Zermelo.
*73·01.
*73·02.
*73·03.
*73·04.
*73·1.
*73·11.
Dem.
*73·12.
*73·13.
Dem.
*73·131.
*73·14.
Dem.
The use of this proposition in proving similarity is very frequent.
*73·141.
*73·142.
Dem.
*73·15.
Dem.
*73·2.
Dem.
The following propositions, down to *73·241, are deduced from preceding propositions of this number just as "" was deduced in *73·2 from *73·1. The proofs are therefore merely indicated by references to the previous propositions of this number which are used.
*73·21.
*73·22.
*73·23.
*73·231.
*73·24.
*73·241.
*73·25.
Dem.
This proposition will be convenient in such cases as the following: Let be a class of relations whose domains are mutually exclusive, i.e. such that no two members of have domains which have a member in common, and suppose we wish to prove that the class of these domains is similar to . The class of domains is , and we have (. Hence we have only to prove (putting in place of the of *73·25) which, in the case supposed, is proved immediately.
*73·26.
Dem.
*73·27.
*73·28.
Dem.
*73·3.
Dem.
This is the reflexive property of similarity. The conditions of significance require that should be a class of some type, but impose no restriction as to the type of class.
*73·301.
Dem.
*73·31.
This proposition shows that similarity is a symmetrical relation.
*73·311.
Dem.
*73·32.
This proposition shows that similarity is a transitive relation. Thus we have now proved that similarity is reflexive, symmetrical, and transitive.
*73·33.
*73·34.
Dem.
*73·35.
Dem.
*73·36.
Dem.
*73·37.
Dem.
*73·4.
*73·41.
This proposition is useful, because it gives a class ( similar to but of higher type. Thus if is a cardinal number, and it is known that in a certain type there are classes having terms, it follows that there will be classes having terms in the next higher type, and therefore in the next type above that, and so on. No corresponding means exist for lowering the type.
*73·42.
Dem.
This proposition gives a means of lowering the type without altering the cardinal number, provided our class is composed wholly of unit classes; for is of the type next below the type of . But when is not composed wholly of unit classes, this construction fails.
*73·43.
*73·44.
Dem.
*73·45.
Dem.
*73·46.
*73·47.
Dem.
*73·48.
The following proposition is used in the theory of double similarity (*111·111).
*73·5.
Dem.
*73·501.
Dem.
*73·51.
Dem.
*73·511.
*73·52.
Dem.
*73·521.
*73·53.
*73·531.
*73·61.
*73·611.
*73·62.
*73·621.
*73·63.
Dem.
The above proposition is used once in connection with cardinal addition (*112·231), and once in connection with cardinal multiplication (*114·561).
The following proposition (*73·69) is a lemma for *73·7.
*73·69.
Dem.
*73·7.
*73·701.
Dem.
*73·71.
*73·72.
Dem.
The following propositions give the proof of the Schröder-Bernstein theorem, namely: If one class is similar to part of another, and the other is similar to part of the one, then the two classes are similar. The proof here given is due to Zermelo[60]. An explanation of the following proof is given in connection with another proof in the summary of *94.
*73·8.
Dem.
*73·801.
Here "Hp*73·8" means "the hypothesis of *73·8."
Dem.
*73·802.
Dem.
*73·81.
Dem.
*73·811.
Dem.
*73·812.
Dem.
*73·82.
Dem.
*73·821.
Dem.
*73·83.
Dem.
*73·84.
Dem.
*73·841.
Dem.
*73·85.
*73·86.
Dem.
*73·87.
Dem.
*73·88.
Dem.
This is the Schröder-Bernstein theorem.