Some consequences of reflection on the approachability ideal

Abstract

Abstract. We study the approachability ideal I[κ + ] in the context of large cardinals and properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I [ℵ ω+1 ] assuming that ℵ ω is a strong limit. In this case we obtain that club many points in ℵ ω+1 of cofinality ℵ n for some n > 1 are approachable assuming the joint reflection of countable families of stationary subsets of ℵ n . This reflection principle holds under MM for all n > 1 and for each n > 1 is equiconsistent with ℵ n being weakly compact in L. This characterizes the structure of the approachability ideal I [ℵ ω+1 ] in models of MM. We also apply our result to show that the Chang conjecture (κ + , κ) (ℵ 2 , ℵ 1 ) fails in models of MM for all singular cardinals κ.

Magidor–Malitz reflection

“…This fact is a combination of PCF related results of Cummings, Foreman, Magidor and Shelah. For a proof, see [16,Section 4].…”

Covering properties and square principles

“…We first recall the definition of a covering matrix. Our definition matches that given by Sharon and Viale in [7]. We note that similar notions existed prior to Viale's work.…”

“…If D is a nice enough covering matrix, then CP(D) and S(D) are equivalent and R(λ, θ) implies both. The following is proved in [7]: Lemma 2.2. Let θ < λ be regular cardinals, and let D be a θ-covering matrix for λ.…”

“…In [11], again using covering matrices, Viale shows that SCH follows from the existence of sufficiently many internally unbounded ℵ 1 -guessing models. In [7], Sharon and Viale use covering matrices to show that This research was undertaken while the author was a Lady Davis Postdoctoral Fellow. The author would like to thank the Lady Davis Fellowship Trust and the Hebrew University of Jerusalem.…”

Pseudo-Prikry sequences

“…The following lemma is a consequence of work of Cummings [3] and a remark of Sharon and Viale [12]. We give a self-contained proof.…”