A weak (?) consequence of determinacy

It is shown that if every real has a sharp and every subset of ω1 is constructible from a real, then there is an inner model with a Woodin cardinal. The following theorem was produced at the AIM meeting “Descriptive Inner Model Theory,” June 02–06, 2014, in a working group whose participants were the authors listed above. The authors would like to thank AIM for their generous hospitality. Theorem 0.1 Assume that every real has a sharp and every subset of ω1 is constructible from a real. Then there is an inner model with a Woodin cardinal. Let C denote the club filter on ω1, i.e., C = {X ⊂ ω1 : ∃C ⊂ ω1 club C ⊂ X}. Lemma 0.2 (Folklore) Assume that every real has a sharp and every subset of ω1 is constructible from a real. Then C is an ultrafilter. 1Cf. http://aimath.org/pastworkshops/innermodel.html