Forcing and Sets of Reals

Abstract

This chapter describes the first advances using Cohen’s method of forcing that involved large cardinals and the first applications of large cardinals in descriptive set theory. Cohen’s creation transformed set theory, and large cardinal hypotheses played an increasingly prominent role as a consequence. §10 discusses the development of forcing, reviews the basic theory, and then focuses on mild extensions and the Levy collapse. §11 is devoted to Solovay’s inspiring result that if there is an inaccessible cardinal, then in an inner model of a forcing extension, every set of reals is Lebesgue measurable. §12 reviews the historical development of descriptive set theory and establish a working context, one in which the classical results are established in §13 through to a delimitation established by Gödel with L. This sets the stage for the further results about large cardinals and projective sets, a major direction of set-theoretic research from the mid-1960’s onwards. §14 describes Solovay’s germinal work on Σ ¹₂ sets that grew out of his Lebesgue measurability result, and his results and conjectures on the definability of sharps. Then §15 describe how Martin used sharps to extend the methods of classical descriptive set theory to the analysis of Σ ¹₃ sets.