Definability in axiomatic set theory II

This chapter discusses the definability in axiomatic set theory. If Zermelo-Fraenkel (ZF) set theory is consistent, so is ZF with the axiom of choice added (ZFC) + generalized continuum hypothesis (GCH) together with the following additional axioms: (1) there exists a nonconstructible real number, (2) every hereditarily-ordinal-definable set is constructible, (3) there is a real number a such that V = L [a], that is, every set is constructible from a, and (4) every constructible cardinal is a true cardinal. The chapter establishes that the set theory ZFC1 is consistent, where ZFC1 is the set theory obtained from ZFC by adding an axiom that asserts the existence of inaccessible cardinals. The chapter assumes the consistency of ZFC1. The chapter presents a general exposition of forcing is given by Solovay.