Let us begin with positive and negative integers. It is obvious on a moment's consideration that +1 and -1 must both be relations, and in fact must be each other's converses. The obvious and sufficient definition is that +1 is the relation of to , and -1 is the relation of to . Generally, if is any inductive number, will be the relation of to (for any ), and will be the relation of to . According to this definition, is a relation which is one-one so long as is a cardinal number (finite or infinite) and is an inductive cardinal number. But is under no circumstances capable of being identified with , which is not a relation, but a class of classes. Indeed, is every bit as distinct from as is. Fractions are more interesting than positive or negative integers. We need fractions for many purposes, but perhaps most obviously for purposes of measurement. My friend and collaborator Dr A. N. Whitehead has developed a theory of fractions specially adapted for their application to measurement, which is set forth in Principia Mathematica.[14] But if all that is needed is to define objects having the required purely mathematical properties, this purpose can be achieved by a simpler method, which we shall here adopt. We shall define the fraction as being that relation which holds between two inductive numbers , when . This definition enables us to prove that is a one-one relation, provided neither or is zero. And of course is the converse relation to . [14]Vol. III. * 300 ff., especially 303. From the above definition it is clear that the fraction is that relation between two integers and which consists in the fact that . This relation, like the relation , is by no means capable of being identified with the inductive cardinal number , because a relation and a class of classes are objects [Pg 64] of utterly different kinds.[15] It will be seen that is always the same relation, whatever inductive number may be; it is, in short, the relation of 0 to any other inductive cardinal. We may call this the zero of rational numbers; it is not, of course, identical with the cardinal number 0. Conversely, the relation is always the same, whatever inductive number may be. There is not any inductive cardinal to correspond to . We may call it "the infinity of rationals." It is an instance of the sort of infinite that is traditional in mathematics, and that is represented by "." This is a totally different sort from the true Cantorian infinite, which we shall consider in our next chapter. The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is not, in actual fact, a very important notion, and we could dispense with it altogether if there were any object in doing so. The Cantorian infinite, on the other hand, is of the greatest and most fundamental importance; the understanding of it opens the way to whole new realms of mathematics and philosophy.