Determinacy

Abstract

The investigation of the determinacy of games is perhaps the most distinctive and intriguing development of modern set theory, and the correlations eventually established with large cardinals the most remarkable and synthetic. Although focused on sets of reals, the subject was to expand across the breadth of set theory from combinatorics and forcing to large cardinals and inner model theory. As a topical and anticipatory conclusion to the present volume this chapter describes this development. §27 discusses the historical beginnings of the study of infinite games and the early work that led to the formulation of determinacy hypotheses. §28 starts with Solovay’s seminal result on the connection with large cardinals and proceeds to develop the combinatorial theory in that direction. §29 and §30 explore the structural consequences of determinacy in descriptive set theory, a direction of investigation first pursued by Moschovakis and Martin. §31 starts the discussion the consistency of determinacy hypotheses, with Martin’s groundbreaking work. Finally, §32 gives a panoramic survey of recent relative consistency results of Martin, Steel, and especially Woodin.