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PROVABLY
$\Delta_1$ GAMES
Published online by Cambridge University Press: 30 October 2020
Abstract
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.
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- © The Association for Symbolic Logic 2020