Axiomatic set theory

Abstract

In this chapter, we present the Zermelo-Fraenkel axioms for set theory, and sketch the justification of them from the Zermelo hierarchy of Chapter 2. The axiom whose status is least clear is the Axiom of Choice. As a result, it has received special attention from mathematicians, and consequences of its truth or falsity have been noted in various parts of mathematics. We will consider some of these. We also develop a theory of infinite cardinal numbers, based on the Axiom of Choice, and say a few words about other systems of axioms which have been proposed.

A choice of axioms is not purely a subjective task. It is usually expected to achieve some definite aim - some specific theorem or theorems are to be derivable from the axioms - and to this extent the problem is exact and objective. But beyond this there are always other important desiderata of a less exact nature: the axioms should not be too numerous, their system is to be as simple and transparent as possible, and each axiom should have an immediate intuitive meaning by which its appropriateness can be judged directly.

John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior [38]