The consistency strength of the free-subset property for ωω

Abstract

A subset X of a structure S is called free in S if ∀x ∈ Xx ∉ S[X − {x}]; here, S[Y] is the substructure of S generated from Y by the functions of S. For κ, λ, μ cardinals, let Frμ(κ, λ) be the assertion:for every structure S with κ ⊂ S which has at most μ functions and relations there is a subset X ⊂ κ free in S of cardinality ≥ λ.We show that Frω(ωω, ω), the free-subset property for ωω, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].In the first section of this paper we prove some combinatorial facts about Frμ(κ, λ); in particular the first cardinal κ such that Frω(κ, ω) is weakly inaccessible or of cofinality ω (1.2). The second section shows that, under Frω(ωω, ω), ωω is measurable in an inner model. For the convenience of readers not acquainted with the core model κ, we first deduce the existence of 0# (2.1) using the inner model L. Then we adapt the proof to the core model and obtain that ωω is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Frω(ωω, ω) over a ground model which contains an ω-sequence of measurable cardinals. We show in §4 that indeed a coherent sequence of Ramsey cardinals suffices. In §3 we obtain such a sequence as an endsegment of a Prikry sequence.