Submitted. PDF

- R. Schindler, S. Uhlenbrock, and W. H. Woodin, “Mice with Finitely many Woodin Cardinals from Optimal Determinacy Hypotheses,” , 2016. (Submitted)
`@article{SUW, author = {R. Schindler and S. Uhlenbrock and W. H. Woodin}, title = {{Mice with Finitely many Woodin Cardinals from Optimal Determinacy Hypotheses}}, note = {Submitted}, year = 2016 }`

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.