Let us illustrate this possibility by an example: Suppose there were exactly nine individuals in the world. (As to what is meant by the word "individual," I must ask the reader to be patient.) Then the inductive cardinals from 0 up to 9 would be such as we expect, but 10 (defined as ) would be the null-class. It will be remembered that may be defined as follows: is the collection of all those classes which have a term such that, when is taken away, there remains a class of terms. Now applying this definition, we see that, in the case supposed, is a class consisting of no classes, i.e. it is the null-class. The same will be true of , or generally of , unless is zero. Thus 10 and all subsequent inductive cardinals will all be identical, since they will all be the null-class. In such a case the inductive cardinals will not form a progression, nor will it be true that no two have the same successor, for 9 and 10 will both be succeeded by the null-class (10 being itself the null-class). It is in order to prevent such arithmetical catastrophes that we require the axiom of infinity. As a matter of fact, so long as we are content with the arithmetic of finite integers, and do not introduce either infinite integers or infinite classes or series of finite integers or ratios, it is possible to obtain all desired results without the axiom of infinity. That is to say, we can deal with the addition, multiplication, [Pg 132] and exponentiation of finite integers and of ratios, but we cannot deal with infinite integers or with irrationals. Thus the theory of the transfinite and the theory of real numbers fails us. How these various results come about must now be explained.