{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T20:16:30Z","timestamp":1775506590435,"version":"3.50.1"},"reference-count":11,"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":11515,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,9]]},"abstract":"If is a countable recursively saturated structure and T<\/jats:italic> is a recursively axiomatizable theory that is consistent with Th(), then it is well known that can be expanded to a recursively saturated model of T<\/jats:italic> [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(\u03c9, +), then can be expanded to a model of first order Peano arithmetic P<\/jats:italic> just in case is recursively saturated (see [3]).<\/jats:p>In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I<\/jats:italic>() consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D<\/jats:italic>(), consists of the degrees of sets S<\/jats:italic> such that is recursive in S<\/jats:italic>-saturated. In general, I<\/jats:italic>() \u2286 D<\/jats:italic>(). Moreover, I<\/jats:italic>() is obviously an \u201cideal\u201d of degrees. For countable structures , D<\/jats:italic>() is \u201cclosed\u201d in the following sense: For any class C<\/jats:italic> \u2286 2\u03c9<\/jats:sup>, if C<\/jats:italic> is co-r.e. in S<\/jats:italic> for some set S<\/jats:italic> such that , then there is some \u03c3 \u2208 C<\/jats:italic> such that . For uncountable structures , we do not know whether D<\/jats:italic>() must be closed.<\/jats:p>","DOI":"10.2307\/2273590","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:00:58Z","timestamp":1146952858000},"page":"587-604","source":"Crossref","is-referenced-by-count":9,"title":["Expansions of models and turing degrees"],"prefix":"10.1017","volume":"47","author":[{"given":"Julia","family":"Knight","sequence":"first","affiliation":[]},{"given":"Mark","family":"Nadel","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043978_ref011","doi-asserted-by":"publisher","DOI":"10.4064\/fm-41-2-203-271"},{"key":"S0022481200043978_ref005","first-page":"612","volume":"45","author":"Nadel","year":"1980","journal-title":"On a problem of MacDowell and Specker"},{"key":"S0022481200043978_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71108-X"},{"key":"S0022481200043978_ref001","first-page":"33","article-title":"\u03a010-classes and degrees of theories","volume":"173","author":"Jockusch","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200043978_ref010","first-page":"135","volume":"43","author":"Simpson","year":"1978","journal-title":"Sets which do not have subsets of every higher degree"},{"key":"S0022481200043978_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1978-0491158-5"},{"key":"S0022481200043978_ref006","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200043978_ref004","first-page":"257","volume-title":"Infinitistic methods","author":"MacDowell","year":"1961"},{"key":"S0022481200043978_ref007","first-page":"183","volume":"43","author":"Schlipf","year":"1978","journal-title":"Model theory and recursive saturation"},{"key":"S0022481200043978_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/005\/0141595"},{"key":"S0022481200043978_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71117-0"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043978","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T20:47:28Z","timestamp":1558730848000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043978\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1982,9]]}},"alternative-id":["S0022481200043978"],"URL":"https:\/\/doi.org\/10.2307\/2273590","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,9]]}}}