The Zermelo—Fraenkel Axioms

Abstract

In this chapter, I develop an axiomatic framework for set theory. For the most part, the axioms will be simple existence assertions about sets, and it may be argued that they are all self-evident ‘truths’ about sets. But why axiomatize set theory in the first place? Well, for one thing, it is well known that set theory provides a unified framework for the whole of pure mathematics, and surely if anything deserves to be put on a sound basis it is such a foundational subject. “But surely,” you say, “the concept of a set is so simple that nothing further need be said. We simply regard any collection of objects as a single entity in its own right, and that provides us with our set theory.” Alas, nothing could be further from the truth. Certainly, the idea of being able to regard any collection of objects as a single entity forms the very core of set theory. But a great deal more needs to be said about this.