Page:Our knowledge of the external world.djvu/213

 are called “inductive,” I shall call the properties to which they are applicable “inductive” properties. Thus an inductive property of numbers is one which is hereditary and belongs to 0.

Taking any one of the natural numbers, say 29, it is easy to see that it must have all inductive properties. For since such properties belong to 0 and are hereditary, they belong to 1; therefore, since they are hereditary, they belong to 2, and so on; by twenty-nine repetitions of such arguments we show that they belong to 29. We may define the “inductive” numbers as all those that possess all inductive properties; they will be the same as what are called the “natural” numbers, i.e. the ordinary finite whole numbers. To all such numbers, proofs by mathematical induction can be validly applied. They are those numbers, we may loosely say, which can be reached from 0 by successive additions of 1; in other words, they are all the numbers that can be reached by counting.

But beyond all these numbers, there are the infinite numbers, and infinite numbers do not have all inductive properties. Such numbers, therefore, may be called non-inductive. All those properties of numbers which are proved by an imaginary step-by-step process from one number to the next are liable to fail when we come to infinite numbers. The first of the infinite numbers has no immediate predecessor, because there is no greatest finite number; thus no succession of steps from one number to the next will ever reach from a finite number to an infinite one, and the step-by-step method of proof fails. This is another reason for the supposed self-contradictions of infinite number. Many of the most familiar properties of numbers, which custom had led people to regard as logically necessary, are in fact only