{"title":"对称扩展的哈托格-林登鲍姆谱","authors":"Calliope Ryan-Smith","doi":"10.1002/malq.202300047","DOIUrl":null,"url":null,"abstract":"<p>We expand the classic result that <span></span><math>\n <semantics>\n <msub>\n <mi>AC</mi>\n <mi>WO</mi>\n </msub>\n <annotation>$\\mathsf {AC}_\\mathsf {WO}$</annotation>\n </semantics></math> is equivalent to the statement “For all <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>ℵ</mi>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <msup>\n <mi>ℵ</mi>\n <mo>∗</mo>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\aleph (X)=\\aleph ^*(X)$</annotation>\n </semantics></math>” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of <span></span><math>\n <semantics>\n <mi>ZF</mi>\n <annotation>$\\mathsf {ZF}$</annotation>\n </semantics></math>, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of <span></span><math>\n <semantics>\n <mi>ZFC</mi>\n <annotation>$\\mathsf {ZFC}$</annotation>\n </semantics></math>. We prove that all such spectra fall into a very rigid pattern.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"210-223"},"PeriodicalIF":0.4000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300047","citationCount":"0","resultStr":"{\"title\":\"The Hartogs–Lindenbaum spectrum of symmetric extensions\",\"authors\":\"Calliope Ryan-Smith\",\"doi\":\"10.1002/malq.202300047\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We expand the classic result that <span></span><math>\\n <semantics>\\n <msub>\\n <mi>AC</mi>\\n <mi>WO</mi>\\n </msub>\\n <annotation>$\\\\mathsf {AC}_\\\\mathsf {WO}$</annotation>\\n </semantics></math> is equivalent to the statement “For all <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>ℵ</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <msup>\\n <mi>ℵ</mi>\\n <mo>∗</mo>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\aleph (X)=\\\\aleph ^*(X)$</annotation>\\n </semantics></math>” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of <span></span><math>\\n <semantics>\\n <mi>ZF</mi>\\n <annotation>$\\\\mathsf {ZF}$</annotation>\\n </semantics></math>, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of <span></span><math>\\n <semantics>\\n <mi>ZFC</mi>\\n <annotation>$\\\\mathsf {ZFC}$</annotation>\\n </semantics></math>. We prove that all such spectra fall into a very rigid pattern.</p>\",\"PeriodicalId\":49864,\"journal\":{\"name\":\"Mathematical Logic Quarterly\",\"volume\":\"70 2\",\"pages\":\"210-223\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-07-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300047\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Logic Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300047\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300047","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
The Hartogs–Lindenbaum spectrum of symmetric extensions
We expand the classic result that is equivalent to the statement “For all , ” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of , and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of . We prove that all such spectra fall into a very rigid pattern.
期刊介绍：
Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.
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