CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS

GUOZHEN SHEN
{"title":"CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS","authors":"GUOZHEN SHEN","doi":"10.1017/jsl.2024.24","DOIUrl":null,"url":null,"abstract":"<p>A partition is finitary if all its members are finite. For a set <span>A</span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathscr {B}(A)$</span></span></img></span></span> denotes the set of all finitary partitions of <span>A</span>. It is shown consistent with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf {ZF}$</span></span></img></span></span> (without the axiom of choice) that there exist an infinite set <span>A</span> and a surjection from <span>A</span> onto <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathscr {B}(A)$</span></span></img></span></span>. On the other hand, we prove in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf {ZF}$</span></span></img></span></span> some theorems concerning <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathscr {B}(A)$</span></span></img></span></span> for infinite sets <span>A</span>, among which are the following: </p><ol><li><p><span>(1)</span> If there is a finitary partition of <span>A</span> without singleton blocks, then there are no surjections from <span>A</span> onto <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathscr {B}(A)$</span></span></img></span></span> and no finite-to-one functions from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathscr {B}(A)$</span></span></img></span></span> to <span>A</span>.</p></li><li><p><span>(2)</span> For all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$n\\in \\omega $</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$|A^n|&lt;|\\mathscr {B}(A)|$</span></span></img></span></span>.</p></li><li><p><span>(3)</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$|\\mathscr {B}(A)|\\neq |\\mathrm {seq}(A)|$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline11.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathrm {seq}(A)$</span></span></span></span> is the set of all finite sequences of elements of <span>A</span>.</p></li></ol><p></p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":"39 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/jsl.2024.24","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract

A partition is finitary if all its members are finite. For a set A, Abstract Image$\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with Abstract Image$\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto Abstract Image$\mathscr {B}(A)$. On the other hand, we prove in Abstract Image$\mathsf {ZF}$ some theorems concerning Abstract Image$\mathscr {B}(A)$ for infinite sets A, among which are the following:

  1. (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto Abstract Image$\mathscr {B}(A)$ and no finite-to-one functions from Abstract Image$\mathscr {B}(A)$ to A.

  2. (2) For all Abstract Image$n\in \omega $, Abstract Image$|A^n|<|\mathscr {B}(A)|$.

  3. (3) Abstract Image$|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where Abstract Image$\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.

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坎托定理对有限分区可能失效
如果一个分区的所有成员都是有限的,那么这个分区就是有限分区。对于一个集合 A,$\mathscr {B}(A)$ 表示 A 的所有有限分割的集合。与 $\mathsf {ZF}$ 一致(没有选择公理),我们证明了存在一个无限集合 A 和一个从 A 到 $\mathscr {B}(A)$ 的投射。另一方面,我们在 $mathsf {ZF}$ 中证明了关于无限集 A 的 $\mathscr {B}(A)$ 的一些定理,其中包括以下定理:(1) 如果A有一个无单子块的有限分割,那么就不存在从A到$\mathscr {B}(A)$ 的投射,也不存在从$\mathscr {B}(A)$ 到A的有限到一的函数。(2) 对于所有 $n\in \omega $,$|A^n|<|\mathscr {B}(A)|$. (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, 其中 $\mathrm {seq}(A)$ 是 A 元素的所有有限序列的集合。
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