Descriptive Set Theory: Projective Sets

Abstract

Publisher Summary This chapter describes classical and effective descriptive set theory, with emphasis mainly on projective sets. The chapter provides an account of the revival in this subject that has taken place in the past 10 years, a revival based on strong set theoretic hypotheses—notably, projective determinacy. In descriptive set theory one restricts one's attention to simple sets of real numbers: sets of simple topological structure or sets that are definable in some simple way. There are three main advantages to such a restricted interest. (1) Many questions that seem unanswerable for arbitrary sets can be answered for sufficiently simple sets. (2) Many questions that have unpleasant answers for arbitrary sets have pleasant answers for sufficiently simple sets. (3) One derives an interesting structural theory of simple sets—a theory of definability. The chapter also presents the main concepts of the classical theory. It deals with the effective, or Kleene, theory and discusses questions in descriptive set theory that the Zermelo–Fraenkel axioms are not sufficient to answer.