ABSTRACT
The next question is: Under what circumstances do two classes have the same number? The answer is, that they have the same number when their terms can be correlated one to one, so that any one term of either corresponds to one and only one term of the other. This requires that there should be some one-one relation whose domain is the one class and whose converse domain is the other class. Thus, for example, if in a community all the men and all the women are married, and polygamy and polyandry are forbidden, the number of men must be the same as the number of women. It might be thought that a one-one relation could not be defined except by reference to the number 1. But this is not the case. A relation is one-one when, if x and x' have the relation in question to y, then x and x' are identical; while if x has the relation in question to y and y' , then y and y' are identical. Thus it is possible, without the notion of unity, to define what is meant by a one-one relation. But in order to provide for the case of two classes which have no terms, it is necessary to modify slightly the above account of what is meant by saying that two classes have the same number. For if there are no terms, the terms cannot be correlated one to one. We must say: Two classes have the same number when, and only when, there is a one-one relation whose domain includes the one class, and which is such that the class of correlates of the terms of the one class is identical with the other class. From this it appears that two classes having no terms have always the same number of terms; for if
we take any one-one relation whatever, its domain includes the null-class, and the class of correlates of the null-class is again the null-class. When two classes have the same number, they are said to be similar.
