If this is true, it follows, of course, that there are many classes of individuals having terms, and that the total number of individuals in the world is not an inductive number. For, by the axiom, there is at least one class having terms, from which it follows that there are many classes of terms and that is not the number of individuals in the world. Since is any inductive number, it follows that the number of individuals in the world must (if our axiom be true) exceed any inductive number. In view of what we found in the preceding chapter, about the possibility of cardinals which are neither inductive nor reflexive, we cannot infer from our axiom that there are at least individuals, unless we assume the multiplicative axiom. But we do know that there are at least classes of classes, since the inductive cardinals are classes of classes, and form a progression if our axiom is true. The way in which the need for this axiom arises may be explained as follows:—One of Peano's assumptions is that no two inductive cardinals have the same successor, i.e. that we shall not have unless , if and are inductive cardinals. In Chapter VIII. we had occasion to use what is virtually the same as the above assumption of Peano's, namely, that, if is an inductive cardinal, [Pg 131] is not equal to . It might be thought that this could be proved. We can prove that, if is an inductive class, and is the number of members of , then is not equal to . This proposition is easily proved by induction, and might be thought to imply the other. But in fact it does not, since there might be no such class as . What it does imply is this: If is an inductive cardinal such that there is at least one class having members, then is not equal to . The axiom of infinity assures us (whether truly or falsely) that there are classes having members, and thus enables us to assert that is not equal to . But without this axiom we should be left with the possibility that and might both be the null-class.