Distinguishing internally club and approachable on an Infinite interval

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-02-21 DOI:10.1112/blms.70013

Hannes Jakob, Maxwell Levine

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引用次数: 0

Abstract

Krueger showed that $PFA$ implies that for all regular $Θ ⩾ ℵ_{2}$ , there are stationarily many ${[H (Θ)]}^{ℵ_{1}}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive $n < ω$ and $Θ ⩾ ℵ_{n + 1}$ , there is a stationary subset of ${[H (Θ)]}^{ℵ_{n}}$ 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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区分内部俱乐部和可接近的无限间隔

Krueger展示了PFA ${\textsf {PFA}}$ 这意味着对于所有规则Θ大于或等于2 $\Theta \geqslant \aleph _2$ ，有平稳的多个[H （Θ）]¹ $[H(\Theta)]^{\aleph _1}$ 他们是俱乐部内部的人，但不是内部的人。从可数的Mahlo基数出发，我们建立了一个模型，在这个模型中，对于所有正的n ＆lt；ω $n<\omega$ 和Θ小于或等于n + 1 $\Theta \geqslant \aleph _{n+1}$ ，存在一个平稳子集[H （Θ）] ～ (n) $[H(\Theta)]^{\aleph _n}$ 由内部俱乐部但内部不可接近的集合组成的。该定理是利用米切尔强迫的一种新变体得到的。这回答了克鲁格的问题。

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