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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:
 3e+60 Determinacy principles 3e+15 Descriptive set theory 3e+45 Inner models, including constructibility, ordinal definability, and core models
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