The axioms of set theory

Abstract

It is probably sensible to read through this chapter fairly quickly, to find out the terminology and notation that we shall use, and then to return later to read it and think about it more carefully. The need for axiomatic set theory Mathematics is written in many languages, such as French, German, Russian, Chinese, and, as in the present case, English. Mathematics needs a particular precision, and within each of these languages, most of mathematics, and all the mathematics that we shall do, is written in the language of sets, using statements and arguments that are based on the grammar and logic of the predicate calculus. In this chapter we introduce the set theory that we shall use. This provides us with a framework in which to work; this framework includes a model for the natural numbers (1, 2, 3,…), together with tools to construct all the other number systems (rational, real and complex) and functions that are the subject of mathematical analysis. The predicate calculus involves rules of grammar for writing ‘well-formed formulae’, and for providing mathematical arguments which use them. Well-formed formulae involve variables, and logical operations such as conjunction ( P and Q ), disjunction ( P or Q (or both)), implication ( P implies Q ), negation (not P ), and quantifiers ‘there exists’ and ‘for all’, together, in our case, with sets and the relation ∈.