# Long Borel Games

## TL;DR

This paper establishes a deep connection between the determinacy of long Borel games of length ω² and the existence of specific fine-structural extender models with Woodin cardinals, linking game theory and set theory.

## Contribution

It proves that Borel games of length ω² are determined if and only if certain advanced extender models with Woodin cardinals exist for all countable ordinals.

## Key findings

- Borel games of length ω² are determined under specific set-theoretic assumptions.
- Existence of fine-structural extender models with Woodin cardinals is equivalent to game determinacy.
- Links between game determinacy and large cardinal hypotheses are established.

## Abstract

It is shown that Borel games of length $\omega^2$ are determined if, and only if, for every countable ordinal $\alpha$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $\alpha$-many iterated powersets above a limit of Woodin cardinals.