{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T12:12:29Z","timestamp":1773144749092,"version":"3.50.1"},"reference-count":11,"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":3571,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,6]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>Let <jats:italic>n<\/jats:italic> be a positive integer. By a <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model we mean an <jats:italic>\u03c9<\/jats:italic>-model which is elementary with respect to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007921_inline1\"\/> formulas. We prove the following <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model version of G\u00f6del's Second Incompleteness Theorem. For any recursively axiomatized theory <jats:italic>S<\/jats:italic> in the language of second order arithmetic, if there exists a <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model of <jats:italic>S<\/jats:italic>, then there exists a <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model of <jats:italic>S<\/jats:italic> + \u201cthere is no countable <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model of <jats:italic>S<\/jats:italic>\u201d. We also prove a <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model version of L\u00f6b's Theorem. As a corollary, we obtain a <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-model which is not a <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub>-model.<\/jats:p>","DOI":"10.2178\/jsl\/1082418545","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:38:52Z","timestamp":1109799532000},"page":"612-616","source":"Crossref","is-referenced-by-count":1,"title":["An incompleteness theorem for <i>\u03b2<\/i><sub><i>n<\/i><\/sub>-models"],"prefix":"10.1017","volume":"69","author":[{"given":"Carl","family":"Mummert","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stephen G.","family":"Simpson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007921_ref011","first-page":"59","volume":"40","author":"Steel","year":"1975","journal-title":"Descending sequences of degrees"},{"key":"S0022481200007921_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01700692"},{"key":"S0022481200007921_ref005","first-page":"363","volume":"41","author":"Friedman","year":"1976","journal-title":"Uniformly defined descending sequences of degrees"},{"key":"S0022481200007921_ref004","unstructured":"Friedman Harvey , Subsystems of Set Theory and Analysis, Ph.D. thesis , Massachusetts Institute of Technology, 1967."},{"key":"S0022481200007921_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"},{"key":"S0022481200007921_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/065"},{"key":"S0022481200007921_ref001","doi-asserted-by":"crossref","unstructured":"Blass Andreas R. , Hirst Jeffry L. , and Simpson Stephen G. , Logical analysis of some theorems of combinatorics and topological dynamics, in [9], 1987, pp. 125\u2013156.","DOI":"10.1090\/conm\/065\/891245"},{"key":"S0022481200007921_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-72-2-175-188"},{"key":"S0022481200007921_ref007","first-page":"115","volume":"20","author":"L\u00f6b","year":"1955","journal-title":"Solution of a problem of Leon Henkln"},{"key":"S0022481200007921_ref008","first-page":"48","volume":"37","author":"Shilleto","year":"1972","journal-title":"Minimum models of analysis"},{"key":"S0022481200007921_ref003","unstructured":"Engstr\u00f6m Fredrik , October 2003, Private communication."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007921","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T21:02:51Z","timestamp":1557176571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007921\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2004,6]]}},"alternative-id":["S0022481200007921"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1082418545","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,6]]}}}