{"title":"自由选择的基本平等","authors":"G. Shen","doi":"10.1215/00294527-2021-0028","DOIUrl":null,"url":null,"abstract":"For a cardinal $\\mathfrak{a}$, let $\\mathrm{fin}(\\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\\mathfrak{a}$ and all natural numbers $n$, \\[ 2^{\\mathrm{fin}(\\mathfrak{a})^n}=2^{[\\mathrm{fin}(\\mathfrak{a})]^n}. \\] On the other hand, it is proved that the following statement is consistent with $\\mathsf{ZF}$: there exists an infinite cardinal $\\mathfrak{a}$ such that \\[ 2^{\\mathrm{fin}(\\mathfrak{a})}<2^{\\mathrm{fin}(\\mathfrak{a})^2}<2^{\\mathrm{fin}(\\mathfrak{a})^3}<\\dots<2^{\\mathrm{fin}(\\mathrm{fin}(\\mathfrak{a}))}. \\]","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2019-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Choice-Free Cardinal Equality\",\"authors\":\"G. Shen\",\"doi\":\"10.1215/00294527-2021-0028\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a cardinal $\\\\mathfrak{a}$, let $\\\\mathrm{fin}(\\\\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\\\\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\\\\mathfrak{a}$ and all natural numbers $n$, \\\\[ 2^{\\\\mathrm{fin}(\\\\mathfrak{a})^n}=2^{[\\\\mathrm{fin}(\\\\mathfrak{a})]^n}. \\\\] On the other hand, it is proved that the following statement is consistent with $\\\\mathsf{ZF}$: there exists an infinite cardinal $\\\\mathfrak{a}$ such that \\\\[ 2^{\\\\mathrm{fin}(\\\\mathfrak{a})}<2^{\\\\mathrm{fin}(\\\\mathfrak{a})^2}<2^{\\\\mathrm{fin}(\\\\mathfrak{a})^3}<\\\\dots<2^{\\\\mathrm{fin}(\\\\mathrm{fin}(\\\\mathfrak{a}))}. \\\\]\",\"PeriodicalId\":51259,\"journal\":{\"name\":\"Notre Dame Journal of Formal Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2019-12-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Notre Dame Journal of Formal Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1215/00294527-2021-0028\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Notre Dame Journal of Formal Logic","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1215/00294527-2021-0028","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\mathfrak{a}$ and all natural numbers $n$, \[ 2^{\mathrm{fin}(\mathfrak{a})^n}=2^{[\mathrm{fin}(\mathfrak{a})]^n}. \] On the other hand, it is proved that the following statement is consistent with $\mathsf{ZF}$: there exists an infinite cardinal $\mathfrak{a}$ such that \[ 2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}(\mathfrak{a})^2}<2^{\mathrm{fin}(\mathfrak{a})^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}. \]
期刊介绍:
The Notre Dame Journal of Formal Logic, founded in 1960, aims to publish high quality and original research papers in philosophical logic, mathematical logic, and related areas, including papers of compelling historical interest. The Journal is also willing to selectively publish expository articles on important current topics of interest as well as book reviews.