{"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":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"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":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12940","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.