{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T06:08:17Z","timestamp":1774591697070,"version":"3.50.1"},"reference-count":4,"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":11334,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,3]]},"abstract":"<jats:p>A model <jats:italic>M<\/jats:italic> (of a countable first order language) is said to be <jats:italic>finitely generated<\/jats:italic> if it is prime over a finite set, namely if there is a finite tuple <jats:italic>\u0101<\/jats:italic> in <jats:italic>M<\/jats:italic> such that (<jats:italic>M, \u0101<\/jats:italic>) is a prime model of its own theory. Similarly, if <jats:italic>A<\/jats:italic> \u2282 <jats:italic>M<\/jats:italic>, then <jats:italic>M<\/jats:italic> is said to be <jats:italic>finitely generated over<\/jats:italic><jats:italic>A<\/jats:italic> if there is finite <jats:italic>\u0101<\/jats:italic> in <jats:italic>M<\/jats:italic> such that <jats:italic>M<\/jats:italic> is prime over <jats:italic>A<\/jats:italic> \u22c3 <jats:italic>\u0101<\/jats:italic>. (Note that if Th(<jats:italic>M<\/jats:italic>) has Skolem functions, then <jats:italic>M<\/jats:italic> being prime over <jats:italic>A<\/jats:italic> is equivalent to <jats:italic>M<\/jats:italic> being generated by <jats:italic>A<\/jats:italic> in the <jats:italic>usual<\/jats:italic> sense, that is, <jats:italic>M<\/jats:italic> is the closure of A under functions of the language.) We show here that if <jats:italic>N<\/jats:italic> is <jats:italic>\u0101<\/jats:italic> model of an <jats:italic>\u03c9<\/jats:italic>-stable theory, <jats:italic>M<\/jats:italic> \u227a <jats:italic>N, M<\/jats:italic> is finitely generated, and <jats:italic>N<\/jats:italic> is finitely generated over <jats:italic>M, then N<\/jats:italic> is finitely generated. A corollary is that any countable model of an <jats:italic>\u03c9<\/jats:italic>-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.<\/jats:p><jats:p>The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite <jats:italic>\u03b1<jats:sub>T<\/jats:sub><\/jats:italic>. However the proof in [3] for the case <jats:italic>\u03b1<jats:sub>T<\/jats:sub><\/jats:italic> finite actually shows the following which does not assume <jats:italic>\u03c9<\/jats:italic>-stability): Let A be atomic over a finite set, tp(<jats:italic>\u0101<\/jats:italic> \/ <jats:italic>A<\/jats:italic>) have finite Cantor-Bendixson rank, and <jats:italic>B<\/jats:italic> be atomic over <jats:italic>A<\/jats:italic> \u22c3 <jats:italic>\u0101<\/jats:italic>. Then <jats:italic>B<\/jats:italic> is atomic over a finite set.<\/jats:p>","DOI":"10.2307\/2273329","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:02:24Z","timestamp":1146938544000},"page":"163-166","source":"Crossref","is-referenced-by-count":2,"title":["A note on finitely generated models"],"prefix":"10.1017","volume":"48","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043437_ref003","volume-title":"Proceedings of the Logic Meeting at Brussels and Mons in June 1980","volume":"23","author":"Pillay","year":"1981"},{"key":"S0022481200043437_ref001","unstructured":"Lascar D. , Ordre de Rudin Keisler et Poids dans les th\u00e9ories stables (to appear)."},{"key":"S0022481200043437_ref002","first-page":"330","volume":"44","author":"Lascar","year":"1979","journal-title":"Introduction to forking"},{"key":"S0022481200043437_ref004","volume-title":"Classification theory and the number of nonisomorphic models","author":"Shelah","year":"1978"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043437","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T15:36:42Z","timestamp":1558712202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043437\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1983,3]]}},"alternative-id":["S0022481200043437"],"URL":"https:\/\/doi.org\/10.2307\/2273329","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,3]]}}}