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.
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