INDEPENDENCE RESULTS IN SET THEORY

Abstract

This chapter discusses independence results in set theory. The chapter explains of the proof that the axiom of choice (AC) and the continuum hypothesis (CH) are independent of the axioms of Zermelo-Fraenkel (Z-F) set theory. Complete proofs will not be presented, but all essential points will be treated and only those details which can be handled by standard methods will be omitted. The latter is an infinite collection of axioms generated by a simply stated rule. In contrast, AC has been accepted by most mathematicians as an axiom not requiring further proof and thus the independence of AC is primarily of technical interest. AC was first explicitly stated by Zermelo in the course of attempting to prove the well-ordering theorem of Cantor. This “naïve” attitude came to an end as these attempts proved fruitless and with the axiomatization of set theory the way was opened to discuss their logical relation to the other axioms.