Some consequences of reflection on the approachability ideal

Assaf Sharon; Matteo Viale

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arXiv · 0804.0775

Some consequences of reflection on the approachability ideal

Assaf Sharon·Matteo Viale

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Abstract

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.

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Assaf Sharon, Matteo Viale. 2008-04-04. Some consequences of reflection on the approachability ideal. https://arxiv.org/abs/0804.0775

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