Transferring compactness

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-30 DOI:10.1112/jlms.12940

Tom Benhamou, Jing Zhang

{"title":"Transferring compactness","authors":"Tom Benhamou, Jing Zhang","doi":"10.1112/jlms.12940","DOIUrl":null,"url":null,"abstract":"We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a strongly inaccessible cardinal <math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math> that is <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-<math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-stationary for all <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ω</mi>\n </mrow>\n <annotation>$n\\in \\omega$</annotation>\n </semantics></math> but not weakly compact. This is in sharp contrast to the situation in the constructible universe <math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math>, where <math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math> being <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$(n+1)$</annotation>\n </semantics></math>-<math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-stationary is equivalent to <math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math> being <math>\n <semantics>\n <msubsup>\n <mi>Π</mi>\n <mi>n</mi>\n <mn>1</mn>\n </msubsup>\n <annotation>$\\mathbf {\\Pi }^1_n$</annotation>\n </semantics></math>-indescribable. We also show that it is consistent that there is a cardinal <math>\n <semantics>\n <mrow>\n <mi>κ</mi>\n <mo>⩽</mo>\n <msup>\n <mn>2</mn>\n <mi>ω</mi>\n </msup>\n </mrow>\n <annotation>$\\kappa \\leqslant 2^\\omega$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <mrow>\n <msub>\n <mi>P</mi>\n <mi>κ</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>λ</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$P_\\kappa (\\lambda)$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-stationary for all <math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n <mo>⩾</mo>\n <mi>κ</mi>\n </mrow>\n <annotation>$\\lambda \\geqslant \\kappa$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ω</mi>\n </mrow>\n <annotation>$n\\in \\omega$</annotation>\n </semantics></math>, answering a question of Sakai.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12940","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12940","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a strongly inaccessible cardinal $κ$ that is $n$ - $d$ -stationary for all $n \in ω$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$ , where $κ$ being $(n + 1)$ - $d$ -stationary is equivalent to $κ$ being $Π_{n}^{1}$ -indescribable. We also show that it is consistent that there is a cardinal $κ ⩽ 2^{ω}$ such that $P_{κ} (λ)$ is $n$ -stationary for all $λ ⩾ κ$ and $n \in ω$ , answering a question of Sakai.

查看原文本刊更多论文

传递紧凑性

我们证明，拉丁强迫技术可以用来把弱不可及但非强极限红心的紧凑性转移到强不可及红心。作为一个应用，相对于大红心的存在，我们构建了一个集合论模型，其中存在一个强不可及红心κ\ $kappa$，对于所有n∈ω\ $n\ in \omega$来说，它是n $n$ - d $d$-稳态的，但不是弱紧凑的。这与可构造宇宙 L $L$ 中的情况形成鲜明对比，在可构造宇宙 L $L$ 中，κ $kappa$ 是 ( n + 1 ) $(n+1)$ - d $d$ - 稳定的等价于 κ $kappa$ 是 Π n 1 $mathbf\ {Pi }^1_n$ - 不可描述的。我们还证明了，对于所有 λ ⩾ κ\ $lambda \geqslant \kappa$ 和 n∈ ω $n\in \omega$ 而言，存在一个红心数 κ ⩽ 2 ω $\kappa \leqslant 2^\omega$ 使得 P κ ( λ ) $P_\kappa (\lambda)$ 是 n $n$ - 稳定的，这一点是一致的，回答了 Sakai 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分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.