1978/12/01 by Leo Harrington · 3 citations
Economics, Econometrics and Finance · Mathematics · #Economic theories and models #Advanced Topology and Set Theory #Mathematical and Theoretical Analysis #Determinacy #Converse #Mathematics #Mathematical proof #Discrete mathematics #Pure mathematics #Algebra over a field #Mathematical analysis
paper · doi:10.2307/2273508
openalex publication_date 1978/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
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 .