Optimal proofs of determinacy II


Itay Neeman





We present a general lemma which allows proving determinacy

from Woodin cardinals. The lemma can be used in many different

settings. As a particular application we prove the determinacy

of sets in $\Game^{(n)}(<\omega^2-\Pi^1_1)$, $n\geq 1$. The

assumption we use to prove $\Game^{(n)}(<\omega^2-\Pi^1_1)$

determinacy is optimal in the base theory of $\mbox{\sf ZFC} +

\mbox{${\mathbf\Pi}^1_n$ determinacy}$.