{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T13:31:38Z","timestamp":1781789498530,"version":"3.54.5"},"reference-count":45,"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":2841,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2006,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove a categoricity transfer theorem for tame abstract elementary classes.<\/jats:p><jats:p><jats:italic>Suppose that<jats:italic>K<\/jats:italic>is a \u03c7-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let<\/jats:italic>\u03bb \u2265 Max{\u03c7, LS(<jats:italic>K<\/jats:italic><jats:sup>+<\/jats:sup>}.<jats:italic>If K is categorical in \u03bb and \u03bb<jats:sup>+<\/jats:sup>, then K is categorical in \u03bb<jats:sup>++<\/jats:sup><\/jats:italic>.<\/jats:p><jats:p>Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:<\/jats:p><jats:p><jats:italic>Suppose K is \u03c7-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let<\/jats:italic>\u03bc<jats:sub>0<\/jats:sub>\u2254 Hanf(<jats:italic>K<\/jats:italic>).<jats:italic>If<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006204_inline1\"\/><jats:italic>and K is categorical in some<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006204_inline2\"\/><jats:italic>then K is categorical in \u03bc for all<\/jats:italic>\u03bc<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006204_inline3\"\/>.<\/jats:p>","DOI":"10.2178\/jsl\/1146620158","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:22:51Z","timestamp":1198081371000},"page":"553-568","source":"Crossref","is-referenced-by-count":33,"title":["Shelah's categoricity conjecture from a successor for tame abstract elementary classes"],"prefix":"10.1017","volume":"71","author":[{"given":"Rami","family":"Grossberg","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Monica","family":"Vandieren","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200006204_ref045","unstructured":"Zilber Boris , Analytic and pseudo-analytic structures, Preprint, http:\/\/www.maths.ox.ac.uk\/~zilber."},{"key":"S0022481200006204_ref043","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(98)00015-3"},{"key":"S0022481200006204_ref038","first-page":"329","article-title":"On what I do not understand (and have something to say), model theory","volume":"51","author":"Shelah","year":"2000","journal-title":"Mathematica Japonica"},{"key":"S0022481200006204_ref036","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1990"},{"key":"S0022481200006204_ref034","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762887"},{"key":"S0022481200006204_ref032","first-page":"241","article-title":"The lazy model-theoretician's guide to stability","volume":"18","author":"Shelah","year":"1975","journal-title":"Logique et Analyse"},{"key":"S0022481200006204_ref031","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757882"},{"key":"S0022481200006204_ref030","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0373874"},{"key":"S0022481200006204_ref029","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90007-0"},{"key":"S0022481200006204_ref037","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(98)00016-5"},{"key":"S0022481200006204_ref028","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"S0022481200006204_ref003","unstructured":"Baldwin John , Kueker David , and VanDieren Monica , Upward stability transfer for tame abstract elementary classes. 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