The Principles of Mathematics (1903)

§ 119

If n be any finite number, the number obtained by adding 1 to n is also finite, and is different from n. Thus beginning with 0 we can form a series of numbers by successive additions of 1. We may definite finite numbers, if we choose, as those numbers that can be obtained from 0 by such steps, and that obey mathematical induction. That is, the class of finite numbers is the class of numbers which is contained in every class s to which belongs 0 and the successor of every number belonging to s, where the successor of a number is the number obtained by adding 1 to the given number. Now a0 is not such a number, since, in virtue of propositions already proved, no such number is similar to a part of itself. Hence also no number greater than a0 is finite according to the new definition. But it is easy to prove that every number less than a0 is finite with the new definition as with the old. Hence the two definitions are equivalent. Thus we may define finite numbers either as those that can be reached by mathematical induction, starting from 0 and increasing by 1 at each step, or as those of classes which are not similar to parts of themselves obtained by taking away single terms. These two definitions are both frequently employed, and it is important to realize that either is a consequence of the other. Both will occupy us much hereafter; for the present it is only intended, without controversy, to set forth the bare outlines of the mathematical theory of finite and infinite, leaving the details to be filled in during the course of the work.(§ 119 ¶ 1)