Before we proceed further it is necessary to make a few remarks about certain ideas of an abstract and logical nature which are of constant occurrence in Pure Mathematics.

In the first place, the reader is probably familiar with the notion of a class. It is unnecessary to discuss here any logical difficulties which may be involved in the notion of a ‘class’: roughly speaking we may say that a class is the aggregate or collection of all the entities or objects which possess a certain property, simple or complex. Thus we have the class of British subjects, or members of Parliament, or positive integers, or real numbers.

Moreover, the reader has probably an idea of what is meant by a finite or infinite class. Thus the class of British subjects is a finite class: the aggregate of all British subjects, past, present, and future, has a finite number $$n$$, though of course we cannot tell at present the actual value of $$n$$. The class of present British subjects, on the other hand, has a number $$n$$ which could be ascertained by counting, were the methods of the census effective enough.

On the other hand the class of positive integers is not finite but infinite. This may be expressed more precisely as follows. If $$n$$ is any positive integer, such as $$1000$$, $$1,000,000$$ or any number we like to think of, then there are more than $$n$$ positive integers. Thus, if the number we think of is $$1,000,000$$, there are obviously at least $$1,000,001$$ positive integers. Similarly the class of rational numbers, or of real numbers, is infinite. It is convenient to express this by saying that there are of positive integers, or rational numbers, or real numbers. But the reader must be careful always to remember that by saying this we mean simply that the class in question has not a finite number of members such as $$1000$$ or $$1,000,000$$.