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 $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I$ 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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