{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,8,12]],"date-time":"2022-08-12T11:51:40Z","timestamp":1660305100481},"reference-count":5,"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":4940,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2000,9]]},"abstract":"<jats:p>In [1], S. Buss introduced the systems of Bounded Arithmetic for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline1\" \/> (<jats:italic>i<\/jats:italic> = 0,1,2,\u2026) which has a close relationship to classes in polynomial hierarchy.<\/jats:p><jats:p>In [4], we defined a very special kind of proof-predicate Prf<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline1\" \/> which gives detailed information on bounds of free variables used in the proof. There we also introduced infinitely many G\u00f6del sentences <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline2\" \/> for Prf<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> (<jats:italic>k<\/jats:italic> = 0, 1, 2, \u2026) and showed that the properties of Prf<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline2\" \/> are closely related to the P \u2260 NP problem. Then we presented many conjectures on Prf<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline2\" \/> which imply P \u2260 NP.<\/jats:p><jats:p>Now in [2], Feferman emphasized that the arithmetization of metamathematics must be carried out intensionally. Bounded Arithmetic is a very interesting case in this sense.<\/jats:p><jats:p>In this paper, we also introduce the usual proof-predicate PRF<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline1\" \/> and infinitely many G\u00f6del sentences <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline3\" \/> for PRF<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup><jats:italic>(k<\/jats:italic>= 0, 1, 2, \u2026). Then we show that (Prf<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline2\" \/><jats:italic>)<\/jats:italic>and (PRF<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline3\" \/>) form a good contrast, this contrast is also closely related to the P \u2260 NP problem, and present more conjectures which imply P \u2260 NP.<\/jats:p><jats:p>As in [4] we define <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline1\" \/> to be the following extension of Buss' original <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline1\" \/>.<\/jats:p><jats:p>(1) We add finitely many function symbols which express polynomial time computable functions to Buss' original language of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011944_inline1\" \/>.<\/jats:p><jats:p>(2) All basic axioms on function symbols and \u2264 can be expressed by initial sequents without logical symbols.<\/jats:p>","DOI":"10.2307\/2586703","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:02:41Z","timestamp":1146938561000},"page":"1338-1346","source":"Crossref","is-referenced-by-count":6,"title":["G\u00f6del sentences of bounded arithmetic"],"prefix":"10.1017","volume":"65","author":[{"given":"Gaisi","family":"Takeuti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200011944_ref004","first-page":"247\u2013261","volume-title":"Logic Colloquium '96","author":"Takeuti","year":"1998"},{"key":"S0022481200011944_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90046-2"},{"key":"S0022481200011944_ref001","volume-title":"Bounded arithmetic","author":"Buss","year":"1986"},{"key":"S0022481200011944_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-35-92"},{"key":"S0022481200011944_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90066-2"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200011944","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T20:46:50Z","timestamp":1557348410000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200011944\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,9]]},"references-count":5,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2000,9]]}},"alternative-id":["S0022481200011944"],"URL":"https:\/\/doi.org\/10.2307\/2586703","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,9]]}}}