Stacking mice

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ א3 such that κ and (κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ). Of special interest is 1) with κ = א3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over K‖κ. §0. Introduction. It is a well-known conjecture that the consistency strength of the Proper Forcing Axiom is a supercompact cardinal. In this paper, we show that PFA implies the existence of an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. In fact, we get indiscernibles for a proper class model of this large cardinal property. For the reader interested in determinacy, this is significantly beyond the consistency strength of ADR by theorems ofWoodin. As one might expect from [12], the only two consequences of PFA that are used to prove our lower bound are 20 = א2 (Todorcevic [2] and Velickovic [22]) and the failure of (κ) at all (regular) κ ≥ א2 (Todorcevic [21]). Recall that κ implies (κ ). The papers Schimmerling [11], Schimmerling-Steel [13] and Steel [20] include steps towards measuring the large cardinal consistency strength of the existence of a singular cardinal κ such that κ fails. In Schimmerling [12], it is shown that if κ ≥ 20 ·א2 is a regular cardinal and both κ and (κ) fail, then for every n < ù, there is an inner model with nWoodin cardinals; Steel (unpublished) extended the conclusion to infinitely many Woodin cardinals. Hypotheses about regular cardinals are more to our taste than singular cardinals because we need only apply PFA to posets of cardinality (20) to see 20 = א2 and the failure of (א2) and א2 . For technical reasons, the least κ to which the results of this paper apply is not א2 but א3. The papers mentioned above use the true core model, K . In the theory of K , one first builds the background certified core model, K c , then defines K to be the Mostowski collapse of a certain elementary substructure of K c . Many of the basic Received December 21, 2007. 1Research supported by NSF grant no. 0700047. 2Research supported by DFG grant no. SCHI 484/3-1 and by NSF grant no. DMS-0401312. Parts of the paper were written while the third author was a visiting scholar at the UC Berkeley Math Dept. He would like to thank faculty and staff and in particular John Steel for their hospitality. 3Research supported by NSF grant no. DMS-0401312. c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00