{"title":"不完整性声明的分类","authors":"Henry Towsner, James Walsh","doi":"arxiv-2409.05973","DOIUrl":null,"url":null,"abstract":"For which choices of $X,Y,Z\\in\\{\\Sigma^1_1,\\Pi^1_1\\}$ does no sufficiently\nstrong $X$-sound and $Y$-definable extension theory prove its own\n$Z$-soundness? We give a complete answer, thereby delimiting the\ngeneralizations of G\\\"odel's second incompleteness theorem that hold within\nsecond-order arithmetic.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A classification of incompleteness statements\",\"authors\":\"Henry Towsner, James Walsh\",\"doi\":\"arxiv-2409.05973\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For which choices of $X,Y,Z\\\\in\\\\{\\\\Sigma^1_1,\\\\Pi^1_1\\\\}$ does no sufficiently\\nstrong $X$-sound and $Y$-definable extension theory prove its own\\n$Z$-soundness? We give a complete answer, thereby delimiting the\\ngeneralizations of G\\\\\\\"odel's second incompleteness theorem that hold within\\nsecond-order arithmetic.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"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-2409.05973\",\"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-2409.05973","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently
strong $X$-sound and $Y$-definable extension theory prove its own
$Z$-soundness? We give a complete answer, thereby delimiting the
generalizations of G\"odel's second incompleteness theorem that hold within
second-order arithmetic.
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