Emmanuel Gunther , Miguel Pagano , Pedro Sánchez Terraf , Matías Steinberg
{"title":"对 CTM 强迫方法的正式验证","authors":"Emmanuel Gunther , Miguel Pagano , Pedro Sánchez Terraf , Matías Steinberg","doi":"10.1016/j.apal.2024.103413","DOIUrl":null,"url":null,"abstract":"<div><p>We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model <em>M</em> of <em>ZFC</em>, of generic extensions satisfying <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mo>¬</mo><mrow><mi>CH</mi></mrow></math></span> and <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mrow><mi>CH</mi></mrow></math></span>. Moreover, let <span><math><mi>R</mi></math></span> be the set of instances of the Axiom of Replacement. We isolated a 21-element subset <span><math><mi>Ω</mi><mo>⊆</mo><mi>R</mi></math></span> and defined <span><math><mi>F</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> such that for every <span><math><mi>Φ</mi><mo>⊆</mo><mi>R</mi></math></span> and <em>M</em>-generic <em>G</em>, <span><math><mi>M</mi><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>F</mi><mtext>“</mtext><mi>Φ</mi><mo>∪</mo><mi>Ω</mi></math></span> implies <span><math><mi>M</mi><mo>[</mo><mi>G</mi><mo>]</mo><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>Φ</mi><mo>∪</mo><mo>{</mo><mo>¬</mo><mrow><mi>CH</mi></mrow><mo>}</mo></math></span>, where <em>ZC</em> is Zermelo set theory with Choice.</p><p>To achieve this, we worked in the proof assistant <em>Isabelle</em>, basing our development on the Isabelle/ZF library by L. Paulson and others.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The formal verification of the ctm approach to forcing\",\"authors\":\"Emmanuel Gunther , Miguel Pagano , Pedro Sánchez Terraf , Matías Steinberg\",\"doi\":\"10.1016/j.apal.2024.103413\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model <em>M</em> of <em>ZFC</em>, of generic extensions satisfying <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mo>¬</mo><mrow><mi>CH</mi></mrow></math></span> and <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mrow><mi>CH</mi></mrow></math></span>. Moreover, let <span><math><mi>R</mi></math></span> be the set of instances of the Axiom of Replacement. We isolated a 21-element subset <span><math><mi>Ω</mi><mo>⊆</mo><mi>R</mi></math></span> and defined <span><math><mi>F</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> such that for every <span><math><mi>Φ</mi><mo>⊆</mo><mi>R</mi></math></span> and <em>M</em>-generic <em>G</em>, <span><math><mi>M</mi><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>F</mi><mtext>“</mtext><mi>Φ</mi><mo>∪</mo><mi>Ω</mi></math></span> implies <span><math><mi>M</mi><mo>[</mo><mi>G</mi><mo>]</mo><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>Φ</mi><mo>∪</mo><mo>{</mo><mo>¬</mo><mrow><mi>CH</mi></mrow><mo>}</mo></math></span>, where <em>ZC</em> is Zermelo set theory with Choice.</p><p>To achieve this, we worked in the proof assistant <em>Isabelle</em>, basing our development on the Isabelle/ZF library by L. Paulson and others.</p></div>\",\"PeriodicalId\":50762,\"journal\":{\"name\":\"Annals of Pure and Applied Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-01-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pure and Applied Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0168007224000101\",\"RegionNum\":2,\"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":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007224000101","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
摘要
我们将讨论计算机验证证明的一些要点，即在给定 ZFC 的可数传递集合模型 M 的情况下，构造满足 ZFC+¬CH 和 ZFC+CH 的泛型扩展。此外，让 R 是替换公理的实例集。我们分离出一个 21 元子集 Ω⊆R，并定义了 F:R→R，使得对于每一个 Φ⊆R 和 M 泛函 G，M⊨ZC∪F "Φ∪Ω 意味着 M[G]⊨ZC∪Φ∪{¬CH}，其中 ZC 是带选择的泽梅洛集合论。为了实现这一目标，我们使用了证明助手 Isabelle，以 L. Paulson 等人的 Isabelle/ZF 库为基础进行开发。
The formal verification of the ctm approach to forcing
We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model M of ZFC, of generic extensions satisfying and . Moreover, let be the set of instances of the Axiom of Replacement. We isolated a 21-element subset and defined such that for every and M-generic G, implies , where ZC is Zermelo set theory with Choice.
To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.
期刊介绍：
The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
