{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T00:46:03Z","timestamp":1773881163221,"version":"3.50.1"},"reference-count":6,"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":7041,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>T<\/jats:italic> be a complete O-minimal theory in a language <jats:italic>L<\/jats:italic>. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of <jats:italic>T<\/jats:italic> are definable. Let <jats:italic>L<\/jats:italic>* be <jats:italic>L<\/jats:italic> together with a unary predicate <jats:italic>P<\/jats:italic>. Let <jats:italic>T<\/jats:italic>* be the <jats:italic>L<\/jats:italic>*-theory of all pairs (<jats:italic>N, M<\/jats:italic>), where <jats:italic>M<\/jats:italic> is a Dedekind complete model of <jats:italic>T<\/jats:italic> and <jats:italic>N<\/jats:italic> is an \u217c<jats:italic>M<\/jats:italic>\u217c<jats:sup>+<\/jats:sup>-saturated elementary extension of <jats:italic>N<\/jats:italic> (and <jats:italic>M<\/jats:italic> is the interpretation of <jats:italic>P<\/jats:italic>). Using the definability of types result, we show that <jats:italic>T<\/jats:italic>* is complete and we give a simple set of axioms for <jats:italic>T<\/jats:italic>*. We also show that for every <jats:italic>L<\/jats:italic>*-formula <jats:italic>\u03d5<\/jats:italic>(<jats:bold>x<\/jats:bold>) there is an <jats:italic>L<\/jats:italic>-formula <jats:italic>\u03c8<\/jats:italic>(<jats:bold>x<\/jats:bold>) such that <jats:italic>T<\/jats:italic>* \u22a2 (\u2200<jats:bold>x<\/jats:bold>)(<jats:italic>P<\/jats:italic>(<jats:bold>x<\/jats:bold>) \u2192 (<jats:italic>\u03d5<\/jats:italic>(<jats:bold>x<\/jats:bold>) \u2194 <jats:italic>\u03c8<\/jats:italic>(<jats:bold>x<\/jats:bold>)). This yields the following result:<\/jats:p><jats:p>Let <jats:italic>M<\/jats:italic> be a Dedekind complete model of <jats:italic>T<\/jats:italic>. Let <jats:italic>\u03d5<\/jats:italic>(<jats:bold>x, y<\/jats:bold>) be an <jats:italic>L<\/jats:italic>-formula where <jats:italic>l<\/jats:italic>(<jats:bold>y<\/jats:bold>) \u2013 <jats:italic>k<\/jats:italic>. Let <jats:bold>X<\/jats:bold> = {<jats:italic>X<\/jats:italic> \u2282 <jats:italic>M<jats:sup>k<\/jats:sup><\/jats:italic>: for some <jats:bold>a<\/jats:bold> in an elementary extension <jats:italic>N<\/jats:italic> of <jats:italic>M, X<\/jats:italic> = <jats:italic>\u03d5<\/jats:italic>(<jats:bold>a, y<\/jats:bold>)<jats:sup><jats:italic>N<\/jats:italic><\/jats:sup> \u2229 <jats:italic>M<jats:sup>k<\/jats:sup><\/jats:italic>}. Then there is a formula <jats:italic>\u03c8<\/jats:italic>(<jats:bold>y, z<\/jats:bold>) of <jats:italic>L<\/jats:italic> such that <jats:bold>X<\/jats:bold> = {<jats:italic>\u03c8<\/jats:italic>(<jats:bold>y, b<\/jats:bold>)<jats:sup><jats:italic>M<\/jats:italic><\/jats:sup>: <jats:bold>b<\/jats:bold> in <jats:italic>M<\/jats:italic>}.<\/jats:p>","DOI":"10.2307\/2275712","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:53:29Z","timestamp":1146956009000},"page":"1400-1409","source":"Crossref","is-referenced-by-count":10,"title":["Definability of types, and pairs of O-minimal structures"],"prefix":"10.1017","volume":"59","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019344_ref006","first-page":"709","volume":"51","author":"Pillay","year":"1986","journal-title":"Some remarks on definable equivalence relations in O-minimal structures"},{"key":"S0022481200019344_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833698-1"},{"key":"S0022481200019344_ref003","unstructured":"van den Dries L. and Lewenberg A. , T-convexity and tame extensions, preprint, 1992."},{"key":"S0022481200019344_ref002","unstructured":"van den Dries L. , O-minimality and tame topology , notes circulated in 1992."},{"key":"S0022481200019344_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0084615"},{"key":"S0022481200019344_ref005","first-page":"185","volume":"59","author":"Marker","year":"1994","journal-title":"Definable types in O-minimal theories"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019344","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T00:41:09Z","timestamp":1557880869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019344\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["S0022481200019344"],"URL":"https:\/\/doi.org\/10.2307\/2275712","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12]]}}}