弗雷格理论结构等同于费弗曼体系 T0

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2024-08-22 DOI:10.1016/j.apal.2024.103510

Daichi Hayashi

{"title":"弗雷格理论结构等同于费弗曼体系 T0","authors":"Daichi Hayashi","doi":"10.1016/j.apal.2024.103510","DOIUrl":null,"url":null,"abstract":"<div>Feferman [9] defines an impredicative system <math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math> of explicit mathematics, which is proof-theoretically equivalent to the subsystem <figure><img></figure> of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as <math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math>. To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by [22]. In addition, we give similar results for the system based on Aczel's original Frege structure [1]. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in [24].</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103510"},"PeriodicalIF":0.6000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Theories of Frege structure equivalent to Feferman's system T0\",\"authors\":\"Daichi Hayashi\",\"doi\":\"10.1016/j.apal.2024.103510\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>Feferman [9] defines an impredicative system <math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math> of explicit mathematics, which is proof-theoretically equivalent to the subsystem <figure><img></figure> of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as <math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math>. To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by [22]. In addition, we give similar results for the system based on Aczel's original Frege structure [1]. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in [24].</div>\",\"PeriodicalId\":50762,\"journal\":{\"name\":\"Annals of Pure and Applied Logic\",\"volume\":\"176 1\",\"pages\":\"Article 103510\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-08-22\",\"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/S0168007224001143\",\"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/S0168007224001143","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

费弗曼[9]定义了一个显式数学的redicative系统T0，它在证明理论上等同于二阶算术的子系统。在本文中，我们提出了几个与 T0 具有相同证明论强度的弗雷格结构系统。确切地说，我们首先考虑了最著名的真理论之一克里普克-费弗曼理论，并受[22]的启发，通过两种归纳原则对其进行了扩展。此外，我们还给出了基于 Aczel 原始弗雷格结构[1]的系统的类似结果。最后，我们在坎蒂尼的监督式理论中加入了宇宙的概念，而宇宙的强度是[24]中的一个悬而未决的问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Theories of Frege structure equivalent to Feferman's system T0

Feferman [9] defines an impredicative system $T_{0}$ of explicit mathematics, which is proof-theoretically equivalent to the subsystem

of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as

T_{0}

. To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by [22]. In addition, we give similar results for the system based on Aczel's original Frege structure [1]. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in [24].

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

来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

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.