凯托宁的问题和其他大罪

Assaf Rinot, Zhixing You, Jiachen Yuan
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引用次数: 0

摘要

从强制概念的交换投影系统得到的泛函扩展的交集模型最近重新引起了人们的兴趣,特别是在描述集合论的研究中。在这里,我们证明它提供了一个富有成果的框架,为解决一些有关小红心的紧凑性原则的未决问题打开了大门。举例来说,从合适的假设出发,我们构建了满足 ZFC 和以下任意条件的交集模型:1.有一个弱紧凑的红心$\kappa$携带一个不可分解的超滤波器,然而$\kappa$是不可测的。这回答了克托宁在 20 世纪 70 年代末提出的一个问题。2.对于适当的多类红心$\lambda$,最小的$\lambda$-强紧密红心是奇异的。这回答了巴加利亚和马吉多尔的一个问题,他们只要求两个这样的红心。3.有一个强不可及的红心,它的$C$序列数是奇异红心。这回答了兰比-汉森和第一作者的一个问题。
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Ketonen's question and other cardinal sins
Intersection models of generic extensions obtained from a commutative projection systems of notions of forcing has recently regained interest, especially in the study of descriptive set theory. Here, we show that it provides a fruitful framework that opens the door to solving some open problems concerning compactness principles of small cardinals. To exemplify, from suitable assumptions, we construct intersection models satisfying ZFC and any of the following: 1. There is a weakly compact cardinal $\kappa$ carrying an indecomposable ultrafilter, yet $\kappa$ is not measurable. This answers a question of Ketonen from the late 1970's. 2. For proper class many cardinals $\lambda$, the least $\lambda$-strongly compact cardinal is singular. This answers a question of Bagaria and Magidor who asked for merely two such cardinals. 3. There is a strongly inaccessible cardinal whose $C$-sequence number is a singular cardinal. This answers a question of Lambie-Hanson and the first author.
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