{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T10:01:18Z","timestamp":1698746478359},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7862,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,9]]},"abstract":"<jats:p>In Parikh [71] it is shown that, if <jats:bold>T<\/jats:bold> is an r.e. consistent extension of Peano arithmetic <jats:bold>PA<\/jats:bold>, then, for each primitive recursive function <jats:italic>g<\/jats:italic>, there is a formula \u03c6 of <jats:bold>PA<\/jats:bold> such that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200022398_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>(In the following, Proof <jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(z, \u03c6) and Prov <jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(\u03c6) denote the metalinguistic assertions that <jats:italic>z<\/jats:italic> codes a proof of \u03c6 in <jats:bold>T<\/jats:bold> and that \u03c6 is provable in <jats:bold>T<\/jats:bold> respectively, where Proof<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(z, \u250c\u03c6\u2510) and Prov<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(\u250c\u03c6\u2510) are the formalizations of Proof <jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(z,\u03c6) and Prov<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(\u03c6) respectively in the language of <jats:bold>PA<\/jats:bold>, \u250c\u03c6\u2510 denotes the G\u00f6del number of \u03c6 and \u250c\u03c6\u2510 denotes the corresponding numeral. Also, for typographical reasons, subscripts will not be made boldface.) If <jats:italic>g<\/jats:italic> is a rapidly increasing function, we express (1) by saying that Prov<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>(\u250c\u03c6\u2510) has a <jats:italic>much shorter proof modulo g<\/jats:italic> than \u03c6. Parikh's result is based on the fact that a suitable formula <jats:italic>A(x)<\/jats:italic>, roughly asserting that (1) holds with <jats:italic>x<\/jats:italic> in place of \u03c6, has only provable fixed points. In de Jongh and Montagna [89], this situation is generalized and investigated in a modal context. There, a characterization is given of arithmetical formulas arising from modal formulas of a suitable modal language which have only provable fixed points, and Parikh's result is obtained as a particular case.<\/jats:p>","DOI":"10.2307\/2275435","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:45:35Z","timestamp":1146955535000},"page":"844-863","source":"Crossref","is-referenced-by-count":3,"title":["Polynomially and superexponentially shorter proofs in fragments of arithmetic"],"prefix":"10.1017","volume":"57","author":[{"given":"Franco","family":"Montagna","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200022398_ref001","unstructured":"Bennett J. H. [62], On spectra, Ph.D. thesis, Princeton University, Princeton, New Jersey."},{"key":"S0022481200022398_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1973-0432416-X"},{"key":"S0022481200022398_ref006","volume-title":"Studia Logica","author":"de Jongh"},{"key":"S0022481200022398_ref003","volume-title":"Provable fixed points in I\u22bf0 + \u03a91","author":"Carbone","year":"1990"},{"key":"S0022481200022398_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90017-2"},{"key":"S0022481200022398_ref007","first-page":"319","volume-title":"Logic and algorithmic","volume":"30","author":"Gaifman"},{"key":"S0022481200022398_ref017","unstructured":"Verbrugge L. C. [ta], \u03a3-completeness and bounded arithmetic, preprint, University of Amsterdam, Amsterdam, 1989."},{"key":"S0022481200022398_ref011","first-page":"494","volume":"36","author":"Parikh","journal-title":"Existence and feasibility in arithmetic"},{"key":"S0022481200022398_ref002","volume-title":"Bounded arithmetic","author":"Buss"},{"key":"S0022481200022398_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19880340307"},{"key":"S0022481200022398_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90012-2"},{"key":"S0022481200022398_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF00293433"},{"key":"S0022481200022398_ref014","volume-title":"Self-reference and modal logic","author":"Smory\u0144ski"},{"key":"S0022481200022398_ref010","volume-title":"Sentences undecidable in formalized arithmetic: an exposition of the theory of Kurt G\u00f6del","author":"Mostowski"},{"key":"S0022481200022398_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757006"},{"key":"S0022481200022398_ref016","unstructured":"Verbrugge L. C. [88], Does Solovay's completeness theorem extend to bounded arithmetic, Master's thesis, University of Amsterdam, Amsterdam."},{"key":"S0022481200022398_ref019","first-page":"130","volume":"57","author":"Visser","journal-title":"An inside view of EXP"},{"key":"S0022481200022398_ref020","unstructured":"Visser A. [ta], On the \u03a31-conservativity of \u03a31completeness, preprint, University of Utrecht, Utrecht, 1989."},{"key":"S0022481200022398_ref021","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90066-2"},{"key":"S0022481200022398_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19890350308"},{"key":"S0022481200022398_ref018","first-page":"175","volume-title":"Mathematical logic: proceedings of the Heyting \u201c88 summer school","author":"Visser"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022398","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T21:01:23Z","timestamp":1558040483000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200022398\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,9]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1992,9]]}},"alternative-id":["S0022481200022398"],"URL":"https:\/\/doi.org\/10.2307\/2275435","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,9]]}}}