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- W2043313700 abstract "In this paper we shall prove Theorem 0.1. Suppose there is a singular strong limit cardinal κ such that □ κ fails; then AD holds in L(R) . See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (α ω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project. Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □ κ fails for all uncountable cardinals κ. Thus we get immediately: It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies AD L (ℝ) and that PFA plus a measurable yields an inner model of AD ℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models. In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply AD L (ℝ) . (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of AD L (ℝ) from an “unaugmented” forcing axiom." @default.
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- W2043313700 date "2005-12-01" @default.
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- W2043313700 title "PFA implies AD<sup><i>L</i>(ℝ)</sup>" @default.
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- W2043313700 doi "https://doi.org/10.2178/jsl/1129642125" @default.
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