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ON WINNING STRATEGIES FOR $F_\sigma $ GAMES

Part of: Set theory

Published online by Cambridge University Press:  26 August 2025

JUAN P. AGUILERA  and
ROBERT LUBARSKY
JUAN P. AGUILERA
Affiliation:
KURT GÖDEL RESEARCH CENTER INSTITUTE OF MATHEMATICS UNIVERSITY OF VIENNA KOLINGASSE 14, 1090 VIENNA AUSTRIA AND INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRASSE 8–10, 1040 VIENNA AUSTRIA AND DEPARTMENT OF MATHEMATICS UNIVERSITY OF GHENT KRIJGSLAAN 281-S8, B9000 GHENT BELGIUM E-mail: aguilera@logic.at
ROBERT LUBARSKY*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES FLORIDA ATLANTIC UNIVERSITY 777 GLADES ROAD, BOCA RATON, FL 33431 USA
*
E-mail: rlubarsk@fau.edu
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Abstract

We prove that every $\Sigma ^0_2$ Gale-Stewart game can be won via a winning strategy $\tau $ which is $\Delta _1$-definable over $L_{\delta }$, the $\delta $th stage of Gödel’s constructible universe, where $\delta = \delta _{\sigma ^1_1}$, strengthening a theorem of Solovay from the 1970s. Moreover, the bound is sharp in the sense that there is a $\Sigma ^0_2$ game with no strategy $\tau $ which is witnessed to be winning by an element of $L_{\delta }$.

Keywords

determinacyhigher recursion theory

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 03E60: Determinacy principles

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Type
Article
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The Journal of Symbolic Logic , First View , pp. 1 - 15
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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