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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:
03E15 Descriptive set theory
03E55 Large cardinals
03E60 Determinacy principles
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