{"title":"作为集合论模型伴侣的二阶算法","authors":"Giorgio Venturi, Matteo Viale","doi":"10.1007/s00153-022-00831-9","DOIUrl":null,"url":null,"abstract":"<div><p>This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a <span>\\(\\Pi _2\\)</span>-property formalized in an appropriate language for second order number theory is forcible from some <span>\\(T\\supseteq \\mathsf {ZFC}+\\)</span><i>large cardinals</i> if and only if it is consistent with the universal fragment of <i>T</i> if and only if it is realized in the model companion of <i>T</i>. In particular we show that the first order theory of <span>\\(H_{\\omega _1}\\)</span> is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for <span>\\(\\Delta _0\\)</span>-properties and for all universally Baire sets of reals. We will extend these results also to the theory of <span>\\(H_{\\aleph _2}\\)</span> in a follow up of this paper.\n\n\n</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00831-9.pdf","citationCount":"3","resultStr":"{\"title\":\"Second order arithmetic as the model companion of set theory\",\"authors\":\"Giorgio Venturi, Matteo Viale\",\"doi\":\"10.1007/s00153-022-00831-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a <span>\\\\(\\\\Pi _2\\\\)</span>-property formalized in an appropriate language for second order number theory is forcible from some <span>\\\\(T\\\\supseteq \\\\mathsf {ZFC}+\\\\)</span><i>large cardinals</i> if and only if it is consistent with the universal fragment of <i>T</i> if and only if it is realized in the model companion of <i>T</i>. In particular we show that the first order theory of <span>\\\\(H_{\\\\omega _1}\\\\)</span> is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for <span>\\\\(\\\\Delta _0\\\\)</span>-properties and for all universally Baire sets of reals. We will extend these results also to the theory of <span>\\\\(H_{\\\\aleph _2}\\\\)</span> in a follow up of this paper.\\n\\n\\n</p></div>\",\"PeriodicalId\":48853,\"journal\":{\"name\":\"Archive for Mathematical Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-06-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00153-022-00831-9.pdf\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archive for Mathematical Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00153-022-00831-9\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Arts and Humanities\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-022-00831-9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
Second order arithmetic as the model companion of set theory
This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a \(\Pi _2\)-property formalized in an appropriate language for second order number theory is forcible from some \(T\supseteq \mathsf {ZFC}+\)large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of \(H_{\omega _1}\) is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for \(\Delta _0\)-properties and for all universally Baire sets of reals. We will extend these results also to the theory of \(H_{\aleph _2}\) in a follow up of this paper.
期刊介绍:
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.