{"title":"Undecidability, incompleteness, and completeness of second-order logic in Coq","authors":"Mark Koch, Dominik Kirst","doi":"10.1145/3497775.3503684","DOIUrl":null,"url":null,"abstract":"We mechanise central metatheoretic results about second-order logic (SOL) using the Coq proof assistant. Concretely, we consider undecidability via many-one reduction from Diophantine equations (Hilbert's tenth problem), incompleteness regarding full semantics via categoricity of second-order Peano arithmetic, and completeness regarding Henkin semantics via translation to mono-sorted first-order logic (FOL). Moreover, this translation is used to transport further characteristic properties of FOL to SOL, namely the compactness and Löwenheim-Skolem theorems.","PeriodicalId":196529,"journal":{"name":"Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3497775.3503684","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We mechanise central metatheoretic results about second-order logic (SOL) using the Coq proof assistant. Concretely, we consider undecidability via many-one reduction from Diophantine equations (Hilbert's tenth problem), incompleteness regarding full semantics via categoricity of second-order Peano arithmetic, and completeness regarding Henkin semantics via translation to mono-sorted first-order logic (FOL). Moreover, this translation is used to transport further characteristic properties of FOL to SOL, namely the compactness and Löwenheim-Skolem theorems.
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