{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T10:45:31Z","timestamp":1648896331793},"reference-count":6,"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":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We define a <jats:italic>rich model<\/jats:italic> to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory <jats:italic>T<\/jats:italic> which has fewer than min(2<jats:sup><jats:italic>\u03bb<\/jats:italic><\/jats:sup>, \u2136<jats:sub>2<\/jats:sub>) rich models of cardinality <jats:italic>\u03bb<\/jats:italic> (<jats:italic>\u03bb<\/jats:italic> &gt; \u2223<jats:italic>T<\/jats:italic>\u2223) is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in \u2135<jats:sub>1<\/jats:sub> and also has a unique countable rich model. We also consider a stronger notion of richness, and use it to characterize superstable theories.<\/jats:p>","DOI":"10.2307\/2274488","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:36:50Z","timestamp":1146940610000},"page":"1292-1298","source":"Crossref","is-referenced-by-count":3,"title":["Rich models"],"prefix":"10.1017","volume":"55","author":[{"given":"Michael H.","family":"Albert","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rami P.","family":"Grossberg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025706_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760644"},{"key":"S0022481200025706_ref006","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah"},{"key":"S0022481200025706_ref003","first-page":"241","article-title":"The lazy model-theoretician's guide to stability theory","volume":"18","author":"Shelah","year":"1975","journal-title":"Logique et Analyse (Nouvelle S\u00e9rie)"},{"key":"S0022481200025706_ref004","first-page":"419","volume-title":"Classification theory (proceedings, Chicago, 1985","volume":"1292","author":"Shelah","year":"1987"},{"key":"S0022481200025706_ref001","first-page":"626","volume":"55","author":"Baldwin","year":"1990","journal-title":"The spectrum of resplendency"},{"key":"S0022481200025706_ref005","first-page":"1431","volume":"54","author":"Shelah","year":"1989","journal-title":"Thenumber of pairwise non-elementarily-embeddable models"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025706","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T16:24:49Z","timestamp":1558196689000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025706\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025706"],"URL":"https:\/\/doi.org\/10.2307\/2274488","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}