{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T18:34:00Z","timestamp":1759775640320},"reference-count":5,"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":3663,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,3]]},"abstract":"<jats:p>The Marker-Steinhorn Theorem (cf. [2] and [3]), says the following. If <jats:italic>T<\/jats:italic> is an o-minimal theory and <jats:italic>M \u227a N<\/jats:italic> is an elementary extension of models of <jats:italic>T<\/jats:italic> such that <jats:italic>M<\/jats:italic> is Dedekind complete in <jats:italic>N<\/jats:italic>, then for every <jats:italic>N<\/jats:italic>-definable subset <jats:italic>X<\/jats:italic> of <jats:italic>N<jats:sup>k<\/jats:sup><\/jats:italic>, the trace <jats:italic>X<\/jats:italic> \u2229 <jats:italic>M<jats:sup>k<\/jats:sup><\/jats:italic> is <jats:italic>M<\/jats:italic>-definable. The original proof in [2] gives an explicit method how to construct a defining formula of <jats:italic>X<\/jats:italic> \u2229 <jats:italic>M<jats:sup>k<\/jats:sup><\/jats:italic> out of a defining formula of <jats:italic>X<\/jats:italic>. A geometric reformulation of the Marker-Steinhorn Theorem is the definability of Hausdorff limits of families of definable sets. An explicit construction of these Hausdorff limits for expansions of the real field has recently been achieved in [1]. Both proofs and also the treatment [3] are technically involved.<\/jats:p><jats:p>Here we give a short algebraic, but not constructive proof, if <jats:italic>T<\/jats:italic> is an expansion of real closed fields. In fact we'll identify the statement of the Theorem with a valuation theoretic property of models of <jats:italic>T<\/jats:italic> (namely condition (\u2020) below). Therefore our proof might be applicable to other elementary classes which expand fields, if a notion of dimension and a reasonable valuation theory are available.<\/jats:p><jats:p>From now on, let <jats:italic>T<\/jats:italic> be an o-minimal expansion of real closed fields. We have to show the following (cf. [2], Th. 2.1. for this formulation). If M is a model of <jats:italic>T<\/jats:italic> and <jats:italic>p<\/jats:italic> is a tame <jats:italic>n<\/jats:italic>-type over <jats:italic>M<\/jats:italic> (i.e., <jats:italic>M<\/jats:italic> is Dedekind complete in <jats:italic>M \u27e8\u1fb1\u27e9<\/jats:italic> := dcl(<jats:italic>M\u1fb1<\/jats:italic>) for some realization \u1fb1 of <jats:italic>p<\/jats:italic>), then <jats:italic>p<\/jats:italic> is a definable type (cf. [4], 11 .b).<\/jats:p>","DOI":"10.2178\/jsl\/1080938828","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T16:26:13Z","timestamp":1109780773000},"page":"91-93","source":"Crossref","is-referenced-by-count":3,"title":["Valuation theoretic content of the Marker-Steinhorn theorem"],"prefix":"10.1017","volume":"69","author":[{"given":"Marcus","family":"Tressl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200008033_ref001","volume-title":"A geometric proof of the definability of Hausdorff limits","author":"Lion","year":"2003"},{"key":"S0022481200008033_ref003","first-page":"1400","volume":"59","author":"Pillay","year":"1994","journal-title":"Definability of types and paires of o-minimal structures"},{"key":"S0022481200008033_ref004","volume-title":"Cours de th\u00e9orie des mod\u00e8les","author":"Poizat","year":"1985"},{"key":"S0022481200008033_ref002","first-page":"185","volume":"59","author":"Marker","year":"1994","journal-title":"Definable types in o-minimal theories"},{"key":"S0022481200008033_ref005","first-page":"74","volume":"60","author":"van den Dries","year":"1995","journal-title":"T-convexity and tame extensions"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200008033","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T17:28:10Z","timestamp":1557163690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200008033\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2004,3]]}},"alternative-id":["S0022481200008033"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1080938828","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,3]]}}}