Some consequences of reflection on the approachability ideal

Sharon, Assaf; Viale, Matteo

doi:10.1090/S0002-9947-10-04976-7

Some consequences of reflection on the approachability ideal
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by Assaf Sharon and Matteo Viale PDF

Trans. Amer. Math. Soc. 362 (2010), 4201-4212 Request permission

Abstract:

We study the approachability ideal $\mathcal {I}[\kappa ^+]$ in the context of large cardinals and properties of the regular cardinals below a singular $\kappa$. As a guiding example consider the approachability ideal $\mathcal {I}[\aleph _{\omega +1}]$ assuming that $\aleph _\omega$ is a strong limit. In this case we obtain that club many points in $\aleph _{\omega +1}$ 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 $\mathsf {MM}$ 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 $\mathcal {I}[\aleph _{\omega +1}]$ in models of $\mathsf {MM}$. We also apply our result to show that the Chang conjecture $(\kappa ^+,\kappa )\twoheadrightarrow (\aleph _2,\aleph _1)$ fails in models of $\mathsf {MM}$ for all singular cardinals $\kappa$.

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