Good Scales and Non-Compactness of Squares

Maxwell Levine; Heike Mildenberger

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arXiv · 2412.16071

Good Scales and Non-Compactness of Squares

Maxwell Levine·Heike Mildenberger

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Abstract

Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $κ$ is a singular strong limit of uncountable cofinality, all scales on $κ$ are good, and $\square^*_δ$ holds for all $δ<κ$, then $\square_κ^*$ holds. In this paper we will present a strongly contrasting result for $\aleph_ω$. We construct a model in which $\square_{\aleph_n}$ holds for all $n<ω$, all scales on $\aleph_ω$ are good, but in which $\square_{\aleph_ω}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ω+1})]^{\aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results.

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BibTeX RIS

Maxwell Levine, Heike Mildenberger. 2026-03-16. Good Scales and Non-Compactness of Squares. https://arxiv.org/abs/2412.16071

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