THE POSITIVE THEORY OF INFINITY 197 



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 o. 



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 o 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 o 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 



