Analytic determinacy and 0^#

Abstract

Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this: Theorem. If analytic games are determined, then x 2 exists for all reals x . This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π 1 1 -determinacy (where α − Π 1 1 is the αth level of the difference hierarchy based on − Π 1 1 see [1]). Martin has also shown that the existence of sharps implies < ω 2 − Π 1 1 -determinacy. Our method also produces the following: Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x . The converse to this theorem had been previously proven by Steel [7], [18]. We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results. For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16]. Throughout this paper we will concern ourselves only with methods for obtaining 0 # (rather than x # for all reals x ). By relativizing our arguments to each real x , one can produce x 2 .