# $F_\sigma$ Games and Reflection in $L(\mathbb{R})$

## TL;DR

This paper establishes a deep connection between the determinacy of certain $F_\sigma$ games of length $2$ and the existence of a specific transitive model of set theory with reflection properties, linking game theory and inner model theory.

## Contribution

It proves that $F_\sigma$ game determinacy of length $2$ is equivalent to the existence of a transitive model of KP + AD with reflection of $1$ facts about the next admissible set.

## Key findings

- Determinacy of $F_\sigma$ games of length $2$ is equivalent to a model existence statement.
- Identifies a specific model of KP + AD with reflection properties.
- Bridges the gap between game determinacy and inner model reflection principles.

## Abstract

It is shown that determinacy of $F_\sigma$ games of length $\omega^2$ is equivalent to the existence of a transitive model of KP + AD which contains the reals and reflects $\Pi_1$ facts about the next admissible set.

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Source: https://tomesphere.com/paper/1906.11762