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SHORTENING CLOPEN GAMES

Part of: Classical measure theory Set theory Game theory

Published online by Cambridge University Press:  08 January 2021

JUAN P. AGUILERA
JUAN P. AGUILERA*
Affiliation:
DEPARTMENT OF MATHEMATICS GHENT UNIVERSITY KRIJGSLAAN 281-S8, 9000 GHENT, BELGIUM and INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRASSE 8–10, 1040 VIENNA, AUSTRIAE-mail: aguilera@logic.at
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Abstract

For every countable wellordering $\alpha $ greater than $\omega $ , it is shown that clopen determinacy for games of length $\alpha $ with moves in $\mathbb {N}$ is equivalent to determinacy for a class of shorter games, but with more complicated payoff. In particular, it is shown that clopen determinacy for games of length $\omega ^2$ is equivalent to $\sigma $ -projective determinacy for games of length $\omega $ and that clopen determinacy for games of length $\omega ^3$ is equivalent to determinacy for games of length $\omega ^2$ in the smallest $\sigma $ -algebra on $\mathbb {R}$ containing all open sets and closed under the real game quantifier.

Keywords

determinacyclopen gameinfinite gamelong gameσ-algebra

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 03E60: Determinacy principles 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets 91A44: Games involving topology or set theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1541 - 1554
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Copyright
© Association for Symbolic Logic 2021

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