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$F_\sigma $ GAMES AND REFLECTION IN $L(\mathbb {R})$

Part of: Computability and recursion theory General logic Set theory

Published online by Cambridge University Press:  21 July 2020

J. P. AGUILERA
J. P. AGUILERA*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF GHENT KRIJGSLAAN 281-S8, 9000 GHENT, BELGIUM INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRAßE 8-10, 1040 VIENNA, AUSTRIA E-mail: aguilera@logic.at
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Abstract

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.

As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$, but weaker than ${\mathsf {AD}} + \Sigma _1$-separation.

Keywords

03D60F∑ gamesdeterminacylong gamesadmissible setreflecting model

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03D60: Computability and recursion theory on ordinals, admissible sets, etc. 03D70: Inductive definability 03E15: Descriptive set theory

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 3 , September 2020 , pp. 1102 - 1123
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Copyright
© The Association for Symbolic Logic 2020

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