PROVABLY $\Delta_1$ GAMES

We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals that(1) if the Continuum Hypothesis holds, then all games of length $\o...

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Bibliographic Details
Published in:The Journal of symbolic logic 2020-09, Vol.85 (3), p.1124-1146
Main Authors: Aguilera, J P, Blue, D W
Format: Article
Language:English
Subjects:
Games
Hypotheses
Set theory
Online Access:Get full text
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Summary:We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals that(1) if the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$-definable from a universally Baire parameter (in first-order or $\Omega $-logic) are determined;(2) all games of length $\omega_1$ with payoff constructible relative to the play are determined; and(3) if the Continuum Hypothesis holds, then there is a model of ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined.
ISSN:0022-4812
1943-5886
DOI:10.1017/jsl.2020.71