{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T21:20:07Z","timestamp":1698268807576},"reference-count":7,"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":14072,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,9]]},"abstract":"<jats:p>This note is concerned with an aspect of the length of proof of formulas <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053007_inline1\" \/> in recursively enumerable theories <jats:italic>T<\/jats:italic> adequate for recursive arithmetic. In particular, we consider the relative length of proof of formulas <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053007_inline1\" \/> in the theories <jats:italic>T<\/jats:italic> and <jats:italic>T<\/jats:italic>(<jats:italic>S<\/jats:italic>), where <jats:italic>F<\/jats:italic> represents an r.e. set <jats:italic>A<\/jats:italic> in <jats:italic>T<\/jats:italic> and <jats:italic>T<\/jats:italic>(<jats:italic>S<\/jats:italic>) is the theory obtained from <jats:italic>T<\/jats:italic> by adjunction, as a new axiom, of a sentence <jats:italic>S<\/jats:italic> undecidable in <jats:italic>T<\/jats:italic>.<\/jats:p><jats:p>Throughout the sequel <jats:italic>T<\/jats:italic> is a consistent, r.e. theory with standard formalization [7] in which all recursive functions of one variable are definable, and in which there is a binary formula <jats:italic>x<\/jats:italic> \u2264 satisfying the well-known conditions [7]:<\/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=\"S0022481200053007_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Here <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053007_inline2\" \/> is the constant term corresponding to the natural number <jats:italic>n<\/jats:italic>. <jats:italic>W<jats:sub>n<\/jats:sub><\/jats:italic> is the <jats:italic>n<\/jats:italic>th r.e. set in a standard enumeration of the r.e. sets. Also, we assume an a priori G\u00f6del numbering of our formalism satisfying the usual conditions, so that all formulas are numbers ab initio.<\/jats:p><jats:p>In the more common applications of the theorem below, if <jats:italic>F<\/jats:italic> is a <jats:italic>k<\/jats:italic>-ary formula of <jats:italic>T<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053007_inline4\" \/> is a natural number that measures in some way the length of the shortest proof of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053007_inline3\" \/> in <jats:italic>T<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2272163","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:36:35Z","timestamp":1146936995000},"page":"398-400","source":"Crossref","is-referenced-by-count":2,"title":["A theorem on shortening the length of proof in formal systems of arithmetic"],"prefix":"10.1017","volume":"40","author":[{"given":"Robert A.","family":"di Paola","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053007_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1971-12696-4"},{"key":"S0022481200053007_ref006","first-page":"82","volume-title":"The undecidable","author":"Godel","year":"1965"},{"key":"S0022481200053007_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1966-0195722-9"},{"key":"S0022481200053007_ref002","first-page":"180","volume":"32","author":"Di Paola","year":"1967","journal-title":"Some theorems on extensions of arithmetic"},{"key":"S0022481200053007_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-35-92"},{"key":"S0022481200053007_ref007","volume-title":"Undecidable theories","author":"Tarski","year":"1953"},{"key":"S0022481200053007_ref003","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1966.18.455"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053007","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:21:20Z","timestamp":1559143280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053007\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200053007"],"URL":"https:\/\/doi.org\/10.2307\/2272163","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}