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arXiv · 2404.15230

Distinguishing Internally Club and Approachable on an Infinite Interval

Hannes Jakob·Maxwell Levine
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Abstract

Krueger showed that PFA implies that for all regular $\Theta \ge \aleph_2$, there are stationarily many $[H(\Theta)]^{\aleph_1}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive $n<\omega$ and $\Theta \ge \aleph_{n+1}$, there is a stationary subset of $[H(\Theta)]^{\aleph_n}$ consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.

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Hannes Jakob, Maxwell Levine. 2024-04-23. Distinguishing Internally Club and Approachable on an Infinite Interval. https://arxiv.org/abs/2404.15230

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