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