{"title":"关于投射红心的说明","authors":"Jiaheng Jin, Guozhen Shen","doi":"arxiv-2408.04287","DOIUrl":null,"url":null,"abstract":"For cardinals $\\mathfrak{a}$ and $\\mathfrak{b}$, we write\n$\\mathfrak{a}=^\\ast\\mathfrak{b}$ if there are sets $A$ and $B$ of cardinalities\n$\\mathfrak{a}$ and $\\mathfrak{b}$, respectively, such that there are partial\nsurjections from $A$ onto $B$ and from $B$ onto $A$. $=^\\ast$-equivalence\nclasses are called surjective cardinals. In this article, we show that\n$\\mathsf{ZF}+\\mathsf{DC}_\\kappa$, where $\\kappa$ is a fixed aleph, cannot prove\nthat surjective cardinals form a cardinal algebra, which gives a negative\nsolution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27,\n165--207 (1984)]. Nevertheless, we show that surjective cardinals form a\n``surjective cardinal algebra'', whose postulates are almost the same with\nthose of a cardinal algebra, except that the refinement postulate is replaced\nby the finite refinement postulate. This yields a smoother proof of the\ncancellation law for surjective cardinals, which states that\n$m\\cdot\\mathfrak{a}=^\\ast m\\cdot\\mathfrak{b}$ implies\n$\\mathfrak{a}=^\\ast\\mathfrak{b}$ for all cardinals $\\mathfrak{a},\\mathfrak{b}$\nand all nonzero natural numbers $m$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on surjective cardinals\",\"authors\":\"Jiaheng Jin, Guozhen Shen\",\"doi\":\"arxiv-2408.04287\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For cardinals $\\\\mathfrak{a}$ and $\\\\mathfrak{b}$, we write\\n$\\\\mathfrak{a}=^\\\\ast\\\\mathfrak{b}$ if there are sets $A$ and $B$ of cardinalities\\n$\\\\mathfrak{a}$ and $\\\\mathfrak{b}$, respectively, such that there are partial\\nsurjections from $A$ onto $B$ and from $B$ onto $A$. $=^\\\\ast$-equivalence\\nclasses are called surjective cardinals. In this article, we show that\\n$\\\\mathsf{ZF}+\\\\mathsf{DC}_\\\\kappa$, where $\\\\kappa$ is a fixed aleph, cannot prove\\nthat surjective cardinals form a cardinal algebra, which gives a negative\\nsolution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27,\\n165--207 (1984)]. Nevertheless, we show that surjective cardinals form a\\n``surjective cardinal algebra'', whose postulates are almost the same with\\nthose of a cardinal algebra, except that the refinement postulate is replaced\\nby the finite refinement postulate. This yields a smoother proof of the\\ncancellation law for surjective cardinals, which states that\\n$m\\\\cdot\\\\mathfrak{a}=^\\\\ast m\\\\cdot\\\\mathfrak{b}$ implies\\n$\\\\mathfrak{a}=^\\\\ast\\\\mathfrak{b}$ for all cardinals $\\\\mathfrak{a},\\\\mathfrak{b}$\\nand all nonzero natural numbers $m$.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.04287\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.04287","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
对于红心$\mathfrak{a}$和$\mathfrak{b}$,如果存在红心分别为$\mathfrak{a}$和$\mathfrak{b}$的集合$A$和$B$,从而存在从$A$到$B$和从$B$到$A$的偏射,我们就写$\mathfrak{a}=^\ast\mathfrak{b}$。$=^\ast$-等价类被称为投射红心。在本文中,我们证明了$\mathsf{ZF}+\mathsf{DC}_\kappa$,其中$\kappa$是一个固定的aleph,不能证明投射红心构成了一个红心代数,这给出了特鲁斯[J. Truss, Ann. Pure Appl. Logic 27,165--207 (1984)]提出的一个问题的否定解答。然而,我们证明了投射红心构成了一个 "投射红心代数",其公设与红心代数的公设几乎相同,只是细化公设被有限细化公设所取代。对于所有的红心数$\mathfrak{a}, \mathfrak{b}$和所有非零自然数$m$来说,这意味着$\mathfrak{a}=^\ast\mathfrak{b}$。
For cardinals $\mathfrak{a}$ and $\mathfrak{b}$, we write
$\mathfrak{a}=^\ast\mathfrak{b}$ if there are sets $A$ and $B$ of cardinalities
$\mathfrak{a}$ and $\mathfrak{b}$, respectively, such that there are partial
surjections from $A$ onto $B$ and from $B$ onto $A$. $=^\ast$-equivalence
classes are called surjective cardinals. In this article, we show that
$\mathsf{ZF}+\mathsf{DC}_\kappa$, where $\kappa$ is a fixed aleph, cannot prove
that surjective cardinals form a cardinal algebra, which gives a negative
solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27,
165--207 (1984)]. Nevertheless, we show that surjective cardinals form a
``surjective cardinal algebra'', whose postulates are almost the same with
those of a cardinal algebra, except that the refinement postulate is replaced
by the finite refinement postulate. This yields a smoother proof of the
cancellation law for surjective cardinals, which states that
$m\cdot\mathfrak{a}=^\ast m\cdot\mathfrak{b}$ implies
$\mathfrak{a}=^\ast\mathfrak{b}$ for all cardinals $\mathfrak{a},\mathfrak{b}$
and all nonzero natural numbers $m$.