Some Consequences of Reflection on the Approachability Ideal

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[אω+1] assuming that אω is a strong limit. In this case we obtain that club many points in אω+1 of cofinality אn for some n > 1 are approachable assuming the joint reflection of countable families of stationary subsets of אn. This reflection principle holds under MM for all n > 1 and for each n > 1 is equiconsistent with אn being weakly compact in L. This characterizes the structure of the approachability ideal I[אω+1] in models of MM. We also apply our result to show that the Chang conjecture (κ+, κ) (א2,א1) fails in models of MM for all singular cardinals κ. 1. The approachability ideal In the course of development of the pcf-theory of possible cofinalities, Shelah has introduced several interesting stationary sets on the successor of a singular cardinal. Among these are the sets of approachable and weakly approachable points in κ, where κ is a singular cardinal. Given A = {aα : α < κ} ⊆ [κ], δ is weakly approachable with respect to A if there is an H unbounded in δ of minimal order type such that {H ∩ γ : γ < δ} is covered by {aα : α < δ} and δ is approachable with respect to A if there is an H unbounded in δ of minimal order type such that {H ∩ γ : γ < δ} ⊆ {aα : α < δ}. Definition 1.1. Let κ be a singular cardinal. S is (weakly) approachable if there is a sequence A = {aα : α < κ} ⊆ [κ] and a club C such that δ is (weakly) approachable with respect to α for all δ ∈ S ∩ C. I[κ] is the ideal generated by approachable sets; I[κ, κ] is the ideal generated by weakly approachable sets. It is clear that I[κ] ⊆ I[κ, κ]. For many of the known applications of approachability, it is irrelevant whether we concentrate on the notion of weak approachability or on the apparently stronger notion of approachability. Moreover in the case that κ is a strong limit and singular, I[κ] = I[κ, κ] (section 3.4 and proposition 3.23 of [3]). For this reason we feel free to concentrate our attention Received by the editors April 16, 2008. 2000 Mathematics Subject Classification. Primary 03E04, 03E55; Secondary 03E65.