On the ultrafilter of closed, unbounded sets

Abstract

Solovay proved in 1967 that the axiom of determinateness implies that the filter C generated by closed and unbounded subsets of ω1 is an ultrafilter. It has long been conjectured that a significant part of the theory of the axiom of determinateness should be provable from the hypothesis that C is an ultrafilter, but even the first step of finding inner models with several measurable cardinals has proved elusive. In this paper we show that such models exist. Much of our proof is a modification of Kunen's proof in [3] of the same conclusion from the existence of a measurable cardinal κ such that 2κ > κ+.Since no proof of Solovay's result seems to have been published, we insert a proof here. We want to show that for any set x ⊂ ω1 there is a closed, unbounded set either contained in or disjoint from x. By the lemma of [4] there is a Turing degree d such that either ω1e Є x for all degrees e ≥T d or ω1e ∉ x for all degrees e ≥T d. By a theorem of Sacks [1], [5] every d-admissible is ω1e for some e ≥T d, so it is enough to show that there is a closed, unbounded set of d-admissibles. Let a ⊂ ω have degree d; then is such a set.