Now it is easy to see that this number is not one of the inductive numbers. If is any inductive number, the number of numbers from 0 to (both included) is ; therefore the total number of inductive numbers is greater than , no matter which of the inductive numbers may be. If we arrange the inductive numbers in a series in order of magnitude, this series has no last term; but if is an inductive number, every series whose field has terms has a last term, as it is easy to prove. Such differences might be multiplied ad lib. Thus the number of inductive numbers is a new number, different from all of them, not possessing all inductive properties. It may happen that 0 has a certain [Pg 78] property, and that if has it so has , and yet that this new number does not have it. The difficulties that so long delayed the theory of infinite numbers were largely due to the fact that some, at least, of the inductive properties were wrongly judged to be such as must belong to all numbers; indeed it was thought that they could not be denied without contradiction. The first step in understanding infinite numbers consists in realising the mistakenness of this view. The most noteworthy and astonishing difference between an inductive number and this new number is that this new number is unchanged by adding 1 or subtracting 1 or doubling or halving or any of a number of other operations which we think of as necessarily making a number larger or smaller. The fact of being not altered by the addition of 1 is used by Cantor for the definition of what he calls "transfinite" cardinal numbers; but for various reasons, some of which will appear as we proceed, it is better to define an infinite cardinal number as one which does not possess all inductive properties, i.e. simply as one which is not an inductive number. Nevertheless, the property of being unchanged by the addition of 1 is a very important one, and we must dwell on it for a time.