Measurable Cardinals

Abstract

This chapter describes the measurable cardinals and provides an introduction to the role of large cardinals in the solution of problems in set theory not solvable by conventional methods. An inaccessible cardinal to prove the consistency of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is described. The consistency is a statement about a set of natural numbers (viz., the set of Gödel numbers of theorems of ZFC). Most of the predicates and operations defined in ZFC through the development of ordinals are absolute. The problems can be settled by large cardinal axioms. The chapter discusses the continuum hypothesis (CH) and the generalized continuum hypothesis (GCH), which show that the measurable cardinals give no information. Almost all the large cardinal properties are preserved by mild Cohen extensions; so these large cardinals do not suffice to settle CH. The existence of a measurable cardinal settles many mathematical problems.