{"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><p>Feferman <span><span>[9]</span></span> defines an impredicative system <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> 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 <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. 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 <span><span>[22]</span></span>. In addition, we give similar results for the system based on Aczel's original Frege structure <span><span>[1]</span></span>. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in <span><span>[24]</span></span>.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007224001143","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
Feferman [9] defines an impredicative system 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 . 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].
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