Stacking mice

Abstract

Abstract 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 ≥ ℵ 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 = ℵ 3 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 c ∥ κ .