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 `a _{0}` 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

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.