PFA implies ADL(R

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 [11] 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(αω < κ), and significantly more work will enable one to drop the hypothesis that κ 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: Corollary 0.2 PFA implies ADL(R). It has been known since the early 90’s that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(R), and that PFA plus a measurable yields an inner model of ADR containing all reals and ordinals. 1 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