{"title":"包含 ZF 的有限非经典理论与包含 ZF 的范畴理论的相对一致性","authors":"Marcoen J. T. F. Cabbolet, Adrian R. D. Mathias","doi":"arxiv-2407.18969","DOIUrl":null,"url":null,"abstract":"Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of\nsets and functions has been introduced as the collection of axioms we have to\naccept if we want a foundational theory for (all of) mathematics that is not\nweaker than ZF, that is finitely axiomatized, and that does not have a\ncountable model (if it has a model at all, that is). Here we prove that T is\nrelatively consistent with ZF. We conclude that this is an important step\ntowards showing that T is an advancement in the foundations of mathematics.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF\",\"authors\":\"Marcoen J. T. F. Cabbolet, Adrian R. D. Mathias\",\"doi\":\"arxiv-2407.18969\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of\\nsets and functions has been introduced as the collection of axioms we have to\\naccept if we want a foundational theory for (all of) mathematics that is not\\nweaker than ZF, that is finitely axiomatized, and that does not have a\\ncountable model (if it has a model at all, that is). Here we prove that T is\\nrelatively consistent with ZF. We conclude that this is an important step\\ntowards showing that T is an advancement in the foundations of mathematics.\",\"PeriodicalId\":501502,\"journal\":{\"name\":\"arXiv - MATH - General Mathematics\",\"volume\":\"18 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - General Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.18969\",\"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 - General Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.18969","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
最近，在《公理 10(2)：119 (2021)一文中，介绍了一个关于集合与函数的非经典一阶理论T，如果我们想要一个不弱于ZF、有限公理化、没有可解释模型（如果它有模型的话）的（所有）数学基础理论，那么T就是我们必须接受的公理集合。在此，我们证明 T 与 ZF 相对一致。我们的结论是，这是朝着证明 T 是数学基础的进步迈出的重要一步。
Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of
sets and functions has been introduced as the collection of axioms we have to
accept if we want a foundational theory for (all of) mathematics that is not
weaker than ZF, that is finitely axiomatized, and that does not have a
countable model (if it has a model at all, that is). Here we prove that T is
relatively consistent with ZF. We conclude that this is an important step
towards showing that T is an advancement in the foundations of mathematics.
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