Ketonen's question and other cardinal sins

Assaf Rinot, Zhixing You, Jiachen Yuan
{"title":"Ketonen's question and other cardinal sins","authors":"Assaf Rinot, Zhixing You, Jiachen Yuan","doi":"arxiv-2408.01547","DOIUrl":null,"url":null,"abstract":"Intersection models of generic extensions obtained from a commutative\nprojection systems of notions of forcing has recently regained interest,\nespecially in the study of descriptive set theory. Here, we show that it\nprovides a fruitful framework that opens the door to solving some open problems\nconcerning compactness principles of small cardinals. To exemplify, from\nsuitable assumptions, we construct intersection models satisfying ZFC and any\nof the following: 1. There is a weakly compact cardinal $\\kappa$ carrying an indecomposable\nultrafilter, yet $\\kappa$ is not measurable. This answers a question of Ketonen\nfrom the late 1970's. 2. For proper class many cardinals $\\lambda$, the least $\\lambda$-strongly\ncompact cardinal is singular. This answers a question of Bagaria and Magidor\nwho asked for merely two such cardinals. 3. There is a strongly inaccessible cardinal whose $C$-sequence number is a\nsingular cardinal. This answers a question of Lambie-Hanson and the first\nauthor.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.01547","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

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.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
凯托宁的问题和其他大罪
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
自引率
0.00%
发文量
0
期刊最新文献
Denotational semantics driven simplicial homology? AC and the Independence of WO in Second-Order Henkin Logic, Part II Positively closed parametrized models Neostability transfers in derivation-like theories Tameness Properties in Multiplicative Valued Difference Fields with Lift and Section
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1