{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:03:46Z","timestamp":1775462626535,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":9781,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,6]]},"abstract":"<jats:p>By Solovay's theorem [16], the modal logic of provability GL gives a complete description of the propositional schemata involving the provability predicate Pr<jats:sub>PA<\/jats:sub>(<jats:italic>x<\/jats:italic>) for Peano arithmetic PA, which are provable in PA. However, many important aspects of provability cannot be fully expressed in terms of Pr<jats:sub>PA<\/jats:sub>(<jats:italic>x<\/jats:italic>). For this reason, many authors have introduced extensions of GL which take account either of Rosser constructions or of other important metamathematical formulas (see, for example, [5], [6], [14], [16], and [19]). In this paper, we concentrate on the modal logic of the provability predicate for finitely axiomatizable subtheories of PA; the interest of this modal logic is based on the following facts. First of all, it provides a modal translation of a very important property of PA, namely the essential reflexiveness. Secondly, in view of Orey's theorem [10] it constitutes a possible approach to the study of interpretability of finite extensions of PA. Indeed, by Orey's theorem PA + <jats:italic>\u03b8<\/jats:italic> is interpretable in PA + <jats:italic>\u03b8<\/jats:italic>\u2032 iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200034423_inline1\"\/> for every <jats:italic>n<\/jats:italic>, and, therefore, relative interpretability of finitely axiomatizable extensions of PA can be expressed by means of the provability predicate for finitely axiomatizable subtheories of PA.<\/jats:p><jats:p>In \u00a71, we introduce three modal logics extending GL and discuss their arithmetical interpretations; \u00a72 deals with Kripke semantics for two of these logics. In \u00a73, a theorem on arithmetical completeness is shown, which characterizes the logic of the provability predicate for finitely axiomatizable subtheories of PA; a uniform version of this theorem is proved in \u00a74.<\/jats:p>","DOI":"10.2307\/2274396","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:21:40Z","timestamp":1146954100000},"page":"494-511","source":"Crossref","is-referenced-by-count":14,"title":["Provability in finite subtheories of PA and relative interpretability: a modal investigation"],"prefix":"10.1017","volume":"52","author":[{"given":"Franco","family":"Montagna","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034423_ref019","first-page":"986","volume":"48","author":"\u0160vejdar","year":"1983","journal-title":"Modal analysis of generalized Rosser sentences"},{"key":"S0022481200034423_ref018","first-page":"789","article-title":"Degrees of interpretability","volume":"19","author":"\u0160vejdar","year":"1978","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S0022481200034423_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8601-8"},{"key":"S0022481200034423_ref010","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19610070710"},{"key":"S0022481200034423_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF00284977"},{"key":"S0022481200034423_ref014","volume-title":"Interpretability","author":"Smory\u0144ski"},{"key":"S0022481200034423_ref008","first-page":"303","article-title":"On the diagonalizable algebra of Peano arithmetic","volume":"15","author":"Montagna","year":"1978","journal-title":"Unione Matematica Italiana: Bollettino B"},{"key":"S0022481200034423_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/2045318"},{"key":"S0022481200034423_ref006","first-page":"667","article-title":"On interpretability in theories containing arithmetic","volume":"22","author":"H\u00e1jek","year":"1981","journal-title":"Commentationes Mathematicae Unhersitatis Carolinae"},{"key":"S0022481200034423_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1979-0539907-7"},{"key":"S0022481200034423_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-35-92"},{"key":"S0022481200034423_ref002","first-page":"191","volume":"47","author":"Boolos","year":"1982","journal-title":"Extremely undecidable sentences"},{"key":"S0022481200034423_ref001","volume-title":"The unprovability of consistency: an essay on modal logic","author":"Boolos","year":"1979"},{"key":"S0022481200034423_ref016","unstructured":"Smory\u0144ski C. , Quantified modal logic and self-reference (to appear)."},{"key":"S0022481200034423_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757006"},{"key":"S0022481200034423_ref013","first-page":"329","volume":"46","author":"Smory\u0144ski","year":"1981","journal-title":"Calculating self-referential statements: Guaspari sentences of the first kind"},{"key":"S0022481200034423_ref012","first-page":"827","volume-title":"Handbook of mathematical logic","author":"Smory\u0144ski","year":"1977"},{"key":"S0022481200034423_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90017-2"},{"key":"S0022481200034423_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF00293433"},{"key":"S0022481200034423_ref020","unstructured":"Visser A. , Aspects of diagonalization and provability, Dissertation, Utrecht, 1981."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034423","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T19:47:57Z","timestamp":1558468077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200034423\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,6]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,6]]}},"alternative-id":["S0022481200034423"],"URL":"https:\/\/doi.org\/10.2307\/2274396","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,6]]}}}