# Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

## TL;DR

This paper establishes that under certain determinacy hypotheses and the absence of specific definable sequences, mice with finitely many Woodin cardinals exist and are iterable, advancing the understanding of the connection between determinacy and inner model theory.

## Contribution

It proves the existence and iterability of mice with finitely many Woodin cardinals from optimal determinacy hypotheses, extending previous results and providing a new determinacy transfer theorem.

## Key findings

- Existence of $M_n^\#$ under combined determinacy assumptions.
- $M_n^\#(x)$ exists and is iterable for all reals $x$.
- Determinacy transfer theorem for arbitrary $n \geq 1$.

## Abstract

We prove the following result which is due to the third author. Let $n \geq 1$. If $\boldsymbol\Pi^1_n$ determinacy and $\Pi^1_{n+1}$ determinacy both hold true and there is no $\boldsymbol\Sigma^1_{n+2}$-definable $\omega_1$-sequence of pairwise distinct reals, then $M_n^\#$ exists and is $\omega_1$-iterable. The proof yields that $\boldsymbol\Pi^1_{n+1}$ determinacy implies that $M_n^\#(x)$ exists and is $\omega_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the statement that $\boldsymbol\Pi^1_{n+1}$ determinacy implies $\Game^{(n)}(<\omega^2 - \boldsymbol\Pi^1_1)$ determinacy.