{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T15:46:22Z","timestamp":1762271182002},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>I show that any sentence of<jats:italic>n<\/jats:italic>th-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically. I set forth an enriched second-order language<jats:italic>L<\/jats:italic>. a sentence<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009269_inline1\" \/>of<jats:italic>L<\/jats:italic>(which is true on the intended interpretation of<jats:italic>L<\/jats:italic>), and a compositionally recursive transformation<jats:italic>Tr<\/jats:italic>defined on formulas of<jats:italic>L<\/jats:italic>, and show that they have the following two properties: (<jats:italic>a<\/jats:italic>) in a universe with at least \u2136<jats:sup><jats:italic>n<\/jats:italic>\u22122<\/jats:sup>objects, any formula of<jats:italic>n<\/jats:italic>th-order (pure or applied) arithmetic can be expressed as a formula of<jats:italic>L<\/jats:italic>, and (<jats:italic>b<\/jats:italic>) for any sentence<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009269_inline2\" \/>of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009269_inline3\" \/>is a second-order sentence containing no arithmetical vocabulary, and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009269_inline4\" \/><\/jats:p>","DOI":"10.2178\/jsl\/1190150304","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:16:54Z","timestamp":1197573414000},"page":"1623-1638","source":"Crossref","is-referenced-by-count":19,"title":["Frege's unofficial arithmetic"],"prefix":"10.1017","volume":"67","author":[{"given":"Agust\u00edn","family":"Rayo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009269_ref023","volume-title":"Frege's conception of numbers as objects","author":"Wright","year":"1983"},{"key":"S0022481200009269_ref020","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0068.00379"},{"key":"S0022481200009269_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF00485187"},{"key":"S0022481200009269_ref015","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198239208.001.0001","volume-title":"Language, thought and logic","author":"Heck","year":"1997"},{"key":"S0022481200009269_ref013","first-page":"243","volume-title":"Proceedings of the Aristotelian Society","volume":"97","author":"Hale","year":"1997"},{"key":"S0022481200009269_ref011","volume":"2","author":"Frege","year":"1893\/1903","journal-title":"Grundgesetze der Arithmetik"},{"key":"S0022481200009269_ref010","volume-title":"Die Grundlagen der Arithmetik","author":"Frege","year":"1884"},{"key":"S0022481200009269_ref008","volume-title":"Logic and arithmetic","author":"Bostock","year":"1979"},{"key":"S0022481200009269_ref006","doi-asserted-by":"publisher","DOI":"10.1093\/mind\/XCIV.375.331"},{"key":"S0022481200009269_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2185003"},{"key":"S0022481200009269_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/2026308"},{"key":"S0022481200009269_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2183530"},{"key":"S0022481200009269_ref001","volume-title":"The Frege reader","author":"Beaney","year":"1997"},{"key":"S0022481200009269_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/2026440"},{"key":"S0022481200009269_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/2998341"},{"key":"S0022481200009269_ref014","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"key":"S0022481200009269_ref024","unstructured":"Wright Crispin [1997], The philosophical significance of Frege's theorem. In Richard Heck, Language Thought and Logic."},{"key":"S0022481200009269_ref019","unstructured":"McGee Vann [2000], Everything, In Gila Sher and Richard Tieszen. Between Logic and Intuition."},{"key":"S0022481200009269_ref022","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511570681"},{"key":"S0022481200009269_ref021","doi-asserted-by":"publisher","DOI":"10.1111\/0029-4624.00288"},{"key":"S0022481200009269_ref003","volume-title":"Philosophy of mathematics","author":"Benacerraf","year":"1983"},{"key":"S0022481200009269_ref007","volume-title":"Logic, logic and logic","author":"Boolos","year":"1998"},{"key":"S0022481200009269_ref009","volume-title":"A subject with no object","author":"Burgess","year":"1997"},{"key":"S0022481200009269_ref012","volume-title":"Notes for Ludwig Darmstaedter","author":"Frege","year":"1919"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009269","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T10:42:54Z","timestamp":1708425774000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009269\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12]]},"references-count":24,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,12]]}},"alternative-id":["S0022481200009269"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150304","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,12]]}}}