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PFA implies \(\text{AD}^{L(\mathbb{R})}\). (English) Zbl 1103.03047
The author proves that if there is a singular strong limit cardinal \(\kappa\) such that \(\square_\kappa\) fails, then AD holds in \(L({\mathbb R})\). Since Todorcevic has shown that if the Proper Forcing Axiom (PFA) holds then \(\square_\kappa\) fails for all uncountable cardinals, the author immediately obtains the result that PFA implies \(\text{AD}^{L(\mathbb R)}\). The proof of the main theorem uses a blend of core model theory and descriptive set theory due to Woodin and called the core model induction technique.

MSC:
03E60 Determinacy principles
03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
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