Compactness versus hugeness at successor cardinals

Sean Cox; Monroe Eskew

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arXiv · 2009.14245

Compactness versus hugeness at successor cardinals

Sean Cox·Monroe Eskew

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Abstract

If $κ$ is regular and $2^{<κ}\leqκ^+$, then the existence of a weakly presaturated ideal on $κ^+$ implies $\square^*_κ$. This partially answers a question of Foreman and Magidor about the approachability ideal on $ω_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

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BibTeX RIS

Sean Cox, Monroe Eskew. 2020-09-29. Compactness versus hugeness at successor cardinals. https://arxiv.org/abs/2009.14245

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