THE definition of cardinal numbers which we gave in Chapter II. was applied in Chapter III. to finite numbers, i.e. to the ordinary natural numbers. To these we gave the name "inductive numbers," because we found that they are to be defined as numbers which obey mathematical induction starting from 0. But we have not yet considered collections which do not have an inductive number of terms, nor have we inquired whether such collections can be said to have a number at all. This is an ancient problem, which has been solved in our own day, chiefly by Georg Cantor. In the present chapter we shall attempt to explain the theory of transfinite or infinite cardinal numbers as it results from a combination of his discoveries with those of Frege on the logical theory of numbers. It cannot be said to be certain that there are in fact any infinite collections in the world. The assumption that there are is what we call the "axiom of infinity." Although various ways suggest themselves by which we might hope to prove this axiom, there is reason to fear that they are all fallacious, and that there is no conclusive logical reason for believing it to be true. At the same time, there is certainly no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections. The practical form of this hypothesis, for our present purposes, is the assumption that, if is any inductive number, is not equal to . Various subtleties arise in identifying this form of our assumption with [Pg 77] the form that asserts the existence of infinite collections; but we will leave these out of account until, in a later chapter, we come to consider the axiom of infinity on its own account. For the present we shall merely assume that, if is an inductive number, is not equal to . This is involved in Peano's assumption that no two inductive numbers have the same successor; for, if , then and have the same successor, namely . Thus we are assuming nothing that was not involved in Peano's primitive propositions. Let us now consider the collection of the inductive numbers themselves. This is a perfectly well-defined class. In the first place, a cardinal number is a set of classes which are all similar to each other and are not similar to anything except each other. We then define as the "inductive numbers" those among cardinals which belong to the posterity of 0 with respect to the relation of to , i.e. those which possess every property possessed by 0 and by the successors of possessors, meaning by the "successor" of the number . Thus the class of "inductive numbers" is perfectly definite. By our general definition of cardinal numbers, the number of terms in the class of inductive numbers is to be defined as "all those classes that are similar to the class of inductive numbers"—i.e. this set of classes is the number of the inductive numbers according to our definitions.