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PROVABLY $\Delta_1$ GAMES

Part of: Set theory

Published online by Cambridge University Press:  30 October 2020

J. P. AGUILERA  and
D. W. BLUE
J. P. AGUILERA
Affiliation:
INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRAßE 8-10, 1040VIENNA, AUSTRIA DEPARTMENT OF MATHEMATICS, GHENT UNIVERSITY KRIJGSLAAN 281-S8, 9000GHENT, BELGIUME-mail: aguilera@logic.at
D. W. BLUE
Affiliation:
HARVARD LOGIC CENTER, HARVARD UNIVERSITY 2 ARROW STREET, CAMBRIDGE, MA02138, USAE-mail: blue@g.harvard.edu
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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. (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. (2) all games of length $\omega_1$ with payoff constructible relative to the play are determined; and

  3. (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.

Keywords

determinacylong gamesΩ-logic

MSC classification

Primary: 03E60: Determinacy principles
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 3 , September 2020 , pp. 1124 - 1146
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Copyright
© The Association for Symbolic Logic 2020

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