The independence results of set theory: An informal exposition

Abstract

This paper is an informal exposition of two significant results in the foundations of set theory: G?del's 1938 proof that the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) are consistent with the axioms of Zermelo-Fraenkel set theory (ZF), and Cohen's 1963 proof of the consistency of the denial of GCH and AC. These theorems amount jointly to the independence of GCH and AC. It is our conviction that these results, and the novel methods they employ, occupy a position like that of the more celebrated incompleteness results for arithmetic. Adequately informed discussion of a broad range of issues in foundations of mathematics and epistemology generally is impossible without some idea of these independence proofs. Moreover, this work sheds retrospective light on much of the metamathematical work that preceded it. Philosophers are therefore obligated to acquire some knowledge of it. While these results are not 'recent' by the standards of a fast-moving discipline like logic, the formidable technicalities involved have kept them inaccessible to non-logicians. We hope to improve the situation. Important details have been relegated to an Appendix, to whose nth entry 'An' refers. Only A16 is strictly necessary for a full grasp of the text.