The Principles of Mathematics (1903)

§ 30

Mathematics requires, so far as I know, only two other primitive propositions, the one that material implication is a relation, the other that ∈ (the relation of a term to a class to which it belongs) is a relation[24]. We can now develop the whole of mathematics without further assumptions or indefinables. Certain propositions in the logic of relations deserve to be mentioned, since they are important, and it might be doubted whether they were capable of formal proof. If u, v be any two classes, there is a relation R the assertion of which between any two terms x and y is equivalent to the assertion that x belongs to u and y to v. If u be any class which is not null, there is a relation which all of its terms have to it, and which holds for no other pairs of terms. If R be any relation, and u be any class contained in the class of referents with respet to R, there is a relation which has u for the class of its referents, and is equivalent to R throughout that class; this relation is the same as R where it holds, but has a more restricted domain. (I use domain as synonymous with class of referents.) From this point onwards, the development of the subject is technical: special types of relations are considered, and special branches of mathematics result.(§ 30 ¶ 1)

§ 30 n. 1. There is a difficulty in regard to this primitive proposition, discussed in §§ 53, 94 below.