{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:53:11Z","timestamp":1759146791676},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":741,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:p>Shepherdson [14] showed that for a discrete ordered ring <jats:italic>I, I<\/jats:italic> is a model of <jats:italic>I Open<\/jats:italic> iff <jats:italic>I<\/jats:italic> is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying <jats:italic>PA<\/jats:italic>. We show that if a real closed ordered field <jats:italic>R<\/jats:italic> has an integer part <jats:italic>I<\/jats:italic> that is a nonstandard model of <jats:italic>PA<\/jats:italic> (or even <jats:italic>I<\/jats:italic>\u03a3<jats:sub>4<\/jats:sub>), then <jats:italic>R<\/jats:italic> must be recursively saturated. In particular, the real closure of <jats:italic>I, RC (I)<\/jats:italic>, is recursively saturated. We also show that if <jats:italic>R<\/jats:italic> is a countable recursively saturated real closed ordered field, then there is an integer part <jats:italic>I<\/jats:italic> such that <jats:italic>R = RC(I)<\/jats:italic> and <jats:italic>I<\/jats:italic> is a nonstandard model of <jats:italic>PA<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1264433906","type":"journal-article","created":{"date-parts":[[2010,1,25]],"date-time":"2010-01-25T15:38:59Z","timestamp":1264433939000},"page":"1-11","source":"Crossref","is-referenced-by-count":9,"title":["Real closed fields and models of Peano arithmetic"],"prefix":"10.1017","volume":"75","author":[{"given":"P.","family":"D'Aquino","sequence":"first","affiliation":[]},{"given":"J. F.","family":"Knight","sequence":"additional","affiliation":[]},{"given":"S.","family":"Starchenko","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002802_ref008","first-page":"641","volume":"58","author":"Mourgues","year":"1993","journal-title":"Every real closed field has an integer part"},{"key":"S0022481200002802_ref011","first-page":"151","article-title":"A guide to the identification of admissible sets above structures","volume":"12","author":"Schlipf","year":"1977","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200002802_ref001","first-page":"531","volume":"41","author":"Barwise","year":"1976","journal-title":"An introduction to recursively saturated and resplendent models"},{"key":"S0022481200002802_ref014","first-page":"79","article-title":"A non-standard model for a free variable fragment of number theory","volume":"12","author":"Shepherdson","year":"1964","journal-title":"Bulletin de l'Academic Polonaise des Sciences"},{"key":"S0022481200002802_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/005\/0141595"},{"key":"S0022481200002802_ref002","first-page":"42","volume-title":"Logic, algebra and arithmetic","volume":"26","author":"Biljakovic","year":"2006"},{"key":"S0022481200002802_ref012","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093870818"},{"key":"S0022481200002802_ref005","volume-title":"Lectures in abstract algebra. Volume III","author":"Jacobson","year":"1975"},{"key":"S0022481200002802_ref003","doi-asserted-by":"publisher","DOI":"10.1051\/ita:2007048"},{"key":"S0022481200002802_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1978-0491158-5"},{"key":"S0022481200002802_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833698-1"},{"key":"S0022481200002802_ref004","first-page":"57","volume-title":"Logic Colloquium '80 (Prague, 1980)","author":"Cegielski","year":"1982"},{"key":"S0022481200002802_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833697-X"},{"key":"S0022481200002802_ref010","volume-title":"Arithmetic, proof theory, and computational complexity","author":"Ressayre","year":"1993"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200002802","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,28]],"date-time":"2019-04-28T20:37:08Z","timestamp":1556483828000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200002802\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,3]]}},"alternative-id":["S0022481200002802"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1264433906","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]}}}