完备原理:heyting算法及其扩展的可证明性研究

Annals of Mathematical Logic Pub Date : 1982-08-01 DOI:10.1016/0003-4843(82)90024-9

Albert Visser

{"title":"完备原理:heyting算法及其扩展的可证明性研究","authors":"Albert Visser","doi":"10.1016/0003-4843(82)90024-9","DOIUrl":null,"url":null,"abstract":"<div>In this paper extensions of HA are studied that prove their own completeness, i.e. they prove A → □ A, where □ is interpreted as provability in the theory itself. Motivation is three-fold: (1) these theories are thought to have some intrinsic interest, (2) they are a tool for producing and studying provability principles, (3) they can be used to proved independence results. Work done in the paper connected with these motivations is respectively: <ul><li>1.(i) A characterization is given of theories proving their own completeness, including an appropriate conservation result.</li><li>2.(ii) Some new provability principles are produced. The provability logic of HA is not a sublogic of the of PA. A provability logic plus completeness theorem is given for a certain intuitionistic extension of HA. De Jongh's theorem for propositional logic is a corollary.</li><li>3.(iii) FP-realizability in Beeson's proof that <math><mtext>∦</mtext><msub><mi></mi><mn><mtext>HA</mtext></mn></msub></math> KLS is replaced by theories proving their own completeness. New consequences are <math><mtext>∦</mtext><msub><mi></mi><mn><mtext>HA</mtext><mtext>+−</mtext><mtext>M</mtext><msub><mi></mi><mn>PR</mn></msub></mn></msub><mtext> </mtext><mtext>KLS</mtext></math>, <math><mtext>∦</mtext><msub><mi></mi><mn><mtext>HA+DNS</mtext></mn></msub><mtext> </mtext><mtext>KLS</mtext></math>.</li></ul></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"22 3","pages":"Pages 263-295"},"PeriodicalIF":0.0000,"publicationDate":"1982-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(82)90024-9","citationCount":"53","resultStr":"{\"title\":\"On the completenes principle: A study of provability in heyting's arithmetic and extensions\",\"authors\":\"Albert Visser\",\"doi\":\"10.1016/0003-4843(82)90024-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>In this paper extensions of HA are studied that prove their own completeness, i.e. they prove A → □ A, where □ is interpreted as provability in the theory itself. Motivation is three-fold: (1) these theories are thought to have some intrinsic interest, (2) they are a tool for producing and studying provability principles, (3) they can be used to proved independence results. Work done in the paper connected with these motivations is respectively: <ul><li>1.(i) A characterization is given of theories proving their own completeness, including an appropriate conservation result.</li><li>2.(ii) Some new provability principles are produced. The provability logic of HA is not a sublogic of the of PA. A provability logic plus completeness theorem is given for a certain intuitionistic extension of HA. De Jongh's theorem for propositional logic is a corollary.</li><li>3.(iii) FP-realizability in Beeson's proof that <math><mtext>∦</mtext><msub><mi></mi><mn><mtext>HA</mtext></mn></msub></math> KLS is replaced by theories proving their own completeness. New consequences are <math><mtext>∦</mtext><msub><mi></mi><mn><mtext>HA</mtext><mtext>+−</mtext><mtext>M</mtext><msub><mi></mi><mn>PR</mn></msub></mn></msub><mtext> </mtext><mtext>KLS</mtext></math>, <math><mtext>∦</mtext><msub><mi></mi><mn><mtext>HA+DNS</mtext></mn></msub><mtext> </mtext><mtext>KLS</mtext></math>.</li></ul></div>\",\"PeriodicalId\":100093,\"journal\":{\"name\":\"Annals of Mathematical Logic\",\"volume\":\"22 3\",\"pages\":\"Pages 263-295\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1982-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/0003-4843(82)90024-9\",\"citationCount\":\"53\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Mathematical Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/0003484382900249\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematical Logic","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/0003484382900249","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 53

摘要

本文研究了证明自身完备性的HA的扩展，即证明了A→□A，其中□被解释为理论本身的可证明性。动机有三个方面:(1)这些理论被认为具有某种内在的利益，(2)它们是产生和研究可证明性原则的工具，(3)它们可用于证明独立性的结果。本文所做的与这些动机有关的工作分别是:1.(i)给出了证明其完备性的理论的一个表征，包括一个适当的守恒结果。2.(ii)给出了一些新的可证明性原理。HA的可证明性逻辑不是PA的子逻辑。对于HA的某一直觉扩展，给出了一个可证性逻辑加完备性定理。3.(iii) Beeson证明∦HA KLS被证明其自身完备性的理论所取代的fp -可实现性。新的结果是∦HA+−MPR KLS，∦HA+DNS KLS。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

On the completenes principle: A study of provability in heyting's arithmetic and extensions

In this paper extensions of HA are studied that prove their own completeness, i.e. they prove A → □ A, where □ is interpreted as provability in the theory itself. Motivation is three-fold: (1) these theories are thought to have some intrinsic interest, (2) they are a tool for producing and studying provability principles, (3) they can be used to proved independence results. Work done in the paper connected with these motivations is respectively:

1.
(i) A characterization is given of theories proving their own completeness, including an appropriate conservation result.
2.
(ii) Some new provability principles are produced. The provability logic of HA is not a sublogic of the of PA. A provability logic plus completeness theorem is given for a certain intuitionistic extension of HA. De Jongh's theorem for propositional logic is a corollary.
3.
(iii) FP-realizability in Beeson's proof that $∦_{HA}$ KLS is replaced by theories proving their own completeness. New consequences are $∦_{HA+−M_{PR}} KLS$ , $∦_{HA+DNS} KLS$ .

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助