Inner Models and Ultrafilters in L(R)

We present a characterization of supercompactness measures for ω1 in L(R), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of δ13 = ω2 with ZFC+AD , and the second proving the uniqueness of the supercompactness measure over Pω1 (λ) in L(R) for λ > δ21. Starting with the work of Steel [13] it became clear that there is a deep connection between inner model theory, particularly for minimal inner models with ω Woodin cardinals, and the study of L(R) under determinacy. The connection centers on the discovery that HOD is a fine structural inner model, in the exact sense of the notion developed previously through the work of [3, 7, 8] up to Θ, and in the more general sense of [14], that puts iteration strategies as well as extenders into the models, at Θ. To be precise, HOD is the direct limit of all countable, iterable inner models with ω Woodin cardinals, together with an iteration strategy for this direct limit. This fact was used extensively by Woodin and Steel, among other things to study measures and ultrafilters: Steel used the directed system to show that for every regular κ < Θ, the ω–club filter over κ is an ultrafilter in L(R). Woodin used the system to show that ω1 is <Θ– supercompact in L(R) (meaning that there is a sequence 〈μλ | λ < Θ〉 ∈ L(R) so that μλ is a supercompactness measure over Pω1(λ) for each λ) and huge to κ for each measurable κ below the largest Suslin cardinal. Here we expand the connection in two ways. We use the directed system to obtain an ultrafilter over [Pω1(λ)] <ω1 , and to prove the uniqueness of the supercompactness measure over Pω1(λ) for λ > δ 2 1. Recall that ω1 is λ–supercompact if there is a fine, normal, countably complete ultrafilter over Pω1(λ). The ultrafilter, or more precisely its characteristic function, is a supercompactness measure over Pω1(λ). Solovay derived the existence of such a measure from the determinacy of infinite games on ordinals below λ. He defined a filter Fλ, essentially the club filter over Pω1(λ), and used determinacy for infinite games on ordinals to show that it is an ultrafilter, and hence a supercompactness measure. For λ up to the largest Suslin cardinal, Harrington–Kechris [4] proved the determinacy of the ordinal games relevant to Solovay’s argument from AD. In L(R), δ1 is the largest Suslin cardinal, and it followed therefore that in L(R) under AD, ω1 is λ–supercompact for each λ < δ 2 1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0094174. 1By Θ and δ1 here and throughout the paper we mean Θ L(R) and (δ1) L(R).