Some consequences of reflection on the approachability ideal

Abstract

We study the approachability ideal I [ κ + ] \mathcal {I}[\kappa ^+] in the context of large cardinals and properties of the regular cardinals below a singular κ \kappa . As a guiding example consider the approachability ideal I [ ℵ ω + 1 ] \mathcal {I}[\aleph _{\omega +1}] assuming that ℵ ω \aleph _\omega is a strong limit. In this case we obtain that club many points in ℵ ω + 1 \aleph _{\omega +1} of cofinality ℵ n \aleph _n for some n > 1 n>1 are approachable assuming the joint reflection of countable families of stationary subsets of ℵ n \aleph _n . This reflection principle holds under M M \mathsf {MM} for all n > 1 n>1 and for each n > 1 n>1 is equiconsistent with ℵ n \aleph _n being weakly compact in L L . This characterizes the structure of the approachability ideal I [ ℵ ω + 1 ] \mathcal {I}[\aleph _{\omega +1}] in models of M M \mathsf {MM} . We also apply our result to show that the Chang conjecture ( κ + , κ ) ↠ ( ℵ 2 , ℵ 1 ) (\kappa ^+,\kappa )\twoheadrightarrow (\aleph _2,\aleph _1) fails in models of M M \mathsf {MM} for all singular cardinals κ \kappa .