{"title":"区分内部俱乐部和可接近的无限间隔","authors":"Hannes Jakob, Maxwell Levine","doi":"10.1112/blms.70013","DOIUrl":null,"url":null,"abstract":"<p>Krueger showed that <span></span><math>\n <semantics>\n <mi>PFA</mi>\n <annotation>${\\textsf {PFA}}$</annotation>\n </semantics></math> implies that for all regular <span></span><math>\n <semantics>\n <mrow>\n <mi>Θ</mi>\n <mo>⩾</mo>\n <msub>\n <mi>ℵ</mi>\n <mn>2</mn>\n </msub>\n </mrow>\n <annotation>$\\Theta \\geqslant \\aleph _2$</annotation>\n </semantics></math>, there are stationarily many <span></span><math>\n <semantics>\n <msup>\n <mrow>\n <mo>[</mo>\n <mi>H</mi>\n <mrow>\n <mo>(</mo>\n <mi>Θ</mi>\n <mo>)</mo>\n </mrow>\n <mo>]</mo>\n </mrow>\n <msub>\n <mi>ℵ</mi>\n <mn>1</mn>\n </msub>\n </msup>\n <annotation>$[H(\\Theta)]^{\\aleph _1}$</annotation>\n </semantics></math> that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo><</mo>\n <mi>ω</mi>\n </mrow>\n <annotation>$n<\\omega$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>Θ</mi>\n <mo>⩾</mo>\n <msub>\n <mi>ℵ</mi>\n <mrow>\n <mi>n</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation>$\\Theta \\geqslant \\aleph _{n+1}$</annotation>\n </semantics></math>, there is a stationary subset of <span></span><math>\n <semantics>\n <msup>\n <mrow>\n <mo>[</mo>\n <mi>H</mi>\n <mrow>\n <mo>(</mo>\n <mi>Θ</mi>\n <mo>)</mo>\n </mrow>\n <mo>]</mo>\n </mrow>\n <msub>\n <mi>ℵ</mi>\n <mi>n</mi>\n </msub>\n </msup>\n <annotation>$[H(\\Theta)]^{\\aleph _n}$</annotation>\n </semantics></math> consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1026-1039"},"PeriodicalIF":0.8000,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70013","citationCount":"0","resultStr":"{\"title\":\"Distinguishing internally club and approachable on an Infinite interval\",\"authors\":\"Hannes Jakob, Maxwell Levine\",\"doi\":\"10.1112/blms.70013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Krueger showed that <span></span><math>\\n <semantics>\\n <mi>PFA</mi>\\n <annotation>${\\\\textsf {PFA}}$</annotation>\\n </semantics></math> implies that for all regular <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Θ</mi>\\n <mo>⩾</mo>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\Theta \\\\geqslant \\\\aleph _2$</annotation>\\n </semantics></math>, there are stationarily many <span></span><math>\\n <semantics>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <mi>H</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>Θ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>]</mo>\\n </mrow>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>1</mn>\\n </msub>\\n </msup>\\n <annotation>$[H(\\\\Theta)]^{\\\\aleph _1}$</annotation>\\n </semantics></math> that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo><</mo>\\n <mi>ω</mi>\\n </mrow>\\n <annotation>$n<\\\\omega$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Θ</mi>\\n <mo>⩾</mo>\\n <msub>\\n <mi>ℵ</mi>\\n <mrow>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation>$\\\\Theta \\\\geqslant \\\\aleph _{n+1}$</annotation>\\n </semantics></math>, there is a stationary subset of <span></span><math>\\n <semantics>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <mi>H</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>Θ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>]</mo>\\n </mrow>\\n <msub>\\n <mi>ℵ</mi>\\n <mi>n</mi>\\n </msub>\\n </msup>\\n <annotation>$[H(\\\\Theta)]^{\\\\aleph _n}$</annotation>\\n </semantics></math> consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 4\",\"pages\":\"1026-1039\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-02-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70013\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.70013\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.70013","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Distinguishing internally club and approachable on an Infinite interval
Krueger showed that implies that for all regular , there are stationarily many that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive and , there is a stationary subset of consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.
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