{"title":"Transferring compactness","authors":"Tom Benhamou, Jing Zhang","doi":"10.1112/jlms.12940","DOIUrl":null,"url":null,"abstract":"<p>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 <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math> that is <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-<span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-stationary for all <span></span><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 <span></span><math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math> being <span></span><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>-<span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-stationary is equivalent to <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math> being <span></span><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 <span></span><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 <span></span><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 <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-stationary for all <span></span><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 <span></span><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.</p>","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 that is --stationary for all but not weakly compact. This is in sharp contrast to the situation in the constructible universe , where being --stationary is equivalent to being -indescribable. We also show that it is consistent that there is a cardinal such that is -stationary for all and , answering a question of Sakai.
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.