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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.

03E60 Determinacy principles
03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
Full Text: DOI
[1] DOI: 10.4310/MRL.1995.v2.n5.a6 · Zbl 0847.03024
[2] Fine structure and iteration trees 3 (1994) · Zbl 0805.03042
[3] DOI: 10.1007/BF02773379 · Zbl 1011.03040
[4] The axiom of determinacy, forcing axioms, and the nonstationary ideal (1999) · Zbl 0954.03046
[5] Contemporary Mathematics 95 pp 209– (1984)
[6] Handbook of set theory
[7] The core model iterability problem 8 (1996) · Zbl 0864.03035
[8] DOI: 10.1016/0168-0072(94)00021-T · Zbl 0821.03023
[9] DOI: 10.1016/0168-0072(93)90037-E · Zbl 0805.03043
[10] Cabal seminar 79–81 pp 107– (1983) · Zbl 0511.00005
[11] The Bulletin of Symbolic Logic 6 (2000)
[12] DOI: 10.1090/S0002-9947-99-02411-3 · Zbl 0928.03059
[13] DOI: 10.1007/s00153-004-0227-1 · Zbl 1063.03038
[14] A weak Dodd-Jensen lemma 64 pp 1285– (1999)
[15] Cabal seminar 79–81 (1983)
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