$F_\sigma $ GAMES AND REFLECTION IN $L(\mathbb {R})$. (21st September 2020)

• Record Type: Journal Article
• Authors: AGUILERA, J. P.
• Is Part Of: Journal of symbolic logic. Volume 85:Number 3(2020)
• Page Start: 1102
• Page End: 1123
• Publication Date: 2020-09-21
• Subjects: 03B30, -- 03D60, -- 03D70, -- 03E15, -- 03D60; F∑ games, -- determinacy, -- long games, -- admissible set, -- reflecting model; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2020.20
• Languages: English
• ISSNs: 0022-4812

Abstract

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.