Limit cardinal

Abstract: Starting with GCH and a _9'3K-hypermeasurable cardinal, a model is produced in which 21 = A+ if A is a successor cardinal and 21 = A++ if A is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing.

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 ω 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.