1. Sets. In order to read anything about our subject, you will have to learn the language that is used in it. I have tried to keep the number of technical terms as small as possible, but there is a certain minimum vocabulary that is essential. Much of it consists of ordinary words used in special senses; this practice has both advantages and disadvantages, but has in any case to be endured since it is now too late to change the language completely. Much of the standard language is taken from the theory of sets, a subject with which we are not concerned for its own sake. The theory of sets is, indeed, an independent branch of mathematics. It has its own basic undefined concepts, subject to various axioms; one of these undefined concepts is the notion of “set” itself.

From an intuitive point of view, however, we may think of a set as being a collection of objects of some kind, called its elements, or members, or points. We say that a set contains its elements, or that the elements belong to the set or simply are in the set. The normal usage of set, as in “a set of dishes” or “a set of the works of Bourbaki,” is fairly close to what we should have in mind, although the second phrase suggests some sort of arrangement of the elements which is irrelevant to the mathematical concept.