# The consistency strength of long projective determinacy

## TL;DR

This paper establishes the exact consistency strength of determinacy for long projective games of length ω^2, linking it to the existence of models with multiple Woodin cardinals.

## Contribution

It proves that ω^2-length projective determinacy implies the existence of models with ω + n Woodin cardinals, advancing understanding of the connection between determinacy and large cardinals.

## Key findings

- Determinacy for ω^2-length projective games implies models with ω + n Woodin cardinals.
- The proof constructs models with determinacy and large cardinals from game hypotheses.
- The methods extend to games with specific payoff conditions, like ext{Game}^ ext{R} oldsymbol ext{Pi}^1_1.

## Abstract

We determine the consistency strength of determinacy for projective games of length $\omega^2$. Our main theorem is that $\boldsymbol\Pi^1_{n+1}$-determinacy for games of length $\omega^2$ implies the existence of a model of set theory with $\omega + n$ Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals $A$ such that $M_n(A)$, the canonical inner model for $n$ Woodin cardinals constructed over $A$, satisfies $A = \mathbb{R}$ and the Axiom of Determinacy. Then we argue how to obtain a model with $\omega + n$ Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length $\omega^2$ with payoff in $\Game^\mathbb{R} \boldsymbol\Pi^1_1$ or with $\sigma$-projective payoff.