Some consequences of reflection on the approachability ideal
Abstract
Description
We study the approachability ideal I[κ^+] in the context of large cardinals properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I[\aleph_{ω+1}] assuming that \aleph_ωis strong limit. In this case we obtain that club many points in \aleph_{ω+1} of cofinality \aleph_n for some n>1 are approachable assuming the joint reflection of countable families of stationary subsets of \aleph_n. This reflection principle holds under Martin's maximum for all n>1 and for each n>1 is equiconsistent with \aleph_n being weakly compact in L. This characterizes the structure of the approachability ideal I[\aleph_{ω+1}] in models of Martin's maximum.
11 pages, updated versions available at the author's webpage
11 pages, updated versions available at the author's webpage