The Journal of Symbolic Logic

Research Article

PFA implies AD L(ℝ)

John R. Steel

University of California at Berkeley, Department of Mathematics, Berkeley, CA 94720-3840, USA, E-mail:

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 (αω < 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 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.

(Received December 20 2003)

(Revised June 14 2005)