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A proof of projective determinacy. (English) Zbl 0668.03021
A cardinal $$\delta$$ is said to be a Woodin cardinal if, for every f: $$\delta$$ $$\to \delta$$, there is a $$\kappa <\delta$$ such that $$\kappa$$ is closed under f and such that there is an elementary embedding j: $$V\to M$$ with critical point $$\kappa$$, where M is transitive and $$V_{(j(f))(\kappa)}\in M$$. The authors show that, for each $$n\in \omega$$, if there is a measurable cardinal larger than n Woodin cardinals, then all $$\Pi^ 1_{n+1}$$ subsets of $$^{\omega}\omega$$ are determined. Using a result of H. Woodin they get as a corollary: if there is a measurable cardinal larger than infinitely many Woodin cardinals, then every subset of $$^{\omega}\omega$$ in L($${\mathcal R})$$ is determined (hence the Axiom of Determinacy holds in the class L($${\mathcal R}))$$.
Reviewer: E.Hartová

##### MSC:
 3e+15 Descriptive set theory 3e+55 Large cardinals 3e+60 Determinacy principles
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##### References:
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