members, it contains
sub-classes—in other words, that there are
ways of selecting some of its members (including the extreme cases where we select all or none); and secondly, that the number of sub-classes contained in a class is always greater than the number of members of the class. Of these two propositions, the first is familiar in the case of finite numbers, and is not hard to extend to infinite numbers. The proof of the second is so simple and so instructive that we shall give it:
In the first place, it is clear that the number of sub-classes of a given class (say
) is at least as great as the number of members, since each member constitutes a sub-class, and we thus have a correlation of all the members with some of the sub-classes. Hence it follows that, if the number of sub-classes is not equal to the number of members, it must be greater. Now it is easy to prove that the number is not equal, by showing that, given any one-one relation whose domain is the members and whose converse domain is contained among the set of sub-classes, there must be at least one sub-class not belonging to the converse domain. The proof is as follows:[21] When a one-one correlation
