{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T22:12:14Z","timestamp":1775513534223,"version":"3.50.1"},"reference-count":2,"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":15533,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1971,9]]},"abstract":"<jats:p>In [2] Vaught showed that if <jats:italic>T<\/jats:italic> is a complete theory formalized in the first-order predicate calculus, then it is not possible for <jats:italic>T<\/jats:italic> to have exactly (up to isomorphism) two countable models. In this note we extend his methods to obtain a theorem which implies the above.<\/jats:p><jats:p>First some definitions. We define <jats:italic>F<jats:sub>n<\/jats:sub>(T)<\/jats:italic> to be the set of well-formed formulas (wffs) in the language of <jats:italic>T<\/jats:italic> whose free variables are among <jats:italic>x<jats:sub>1<\/jats:sub> x<jats:sub>2<\/jats:sub>, \u2026, x<jats:sub>n<\/jats:sub><\/jats:italic>. An n-type of <jats:italic>T<\/jats:italic> is a maximal consistent set of wffs of <jats:italic>F<jats:sub>n<\/jats:sub>(T)<\/jats:italic>; equivalently, a subset <jats:italic>P<\/jats:italic> of <jats:italic>F<jats:sub>n<\/jats:sub>(T)<\/jats:italic> is an <jats:italic>n<\/jats:italic>-type of <jats:italic>T<\/jats:italic> if there is a model <jats:italic>M<\/jats:italic> of <jats:italic>T<\/jats:italic> and elements <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>a<\/jats:italic><jats:sub>2<\/jats:sub>, \u2026, <jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic> of <jats:italic>M<\/jats:italic> such that <jats:italic>M<\/jats:italic> \u22a7 <jats:italic>\u03d5<\/jats:italic>(<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>a<\/jats:italic><jats:sub>2<\/jats:sub>, \u2026, <jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>) for every <jats:italic>\u03d5<\/jats:italic> \u2208 <jats:italic>P<\/jats:italic>. In the latter case we say that \u3008<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>a<\/jats:italic><jats:sub>2<\/jats:sub>, \u2026, <jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>\u3009 ony realizes <jats:italic>P<\/jats:italic> in <jats:italic>M<\/jats:italic>. Every set of wffs of <jats:italic>F<jats:sub>n<\/jats:sub>(T)<\/jats:italic> which is consistent with <jats:italic>T<\/jats:italic> can be extended to an <jats:italic>n<\/jats:italic>-type of <jats:italic>T<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2269951","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:09:50Z","timestamp":1146935390000},"page":"439-440","source":"Crossref","is-referenced-by-count":3,"title":["A note on a theorem of Vaught"],"prefix":"10.1017","volume":"36","author":[{"given":"Joseph G.","family":"Rosenstein","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200082232_ref002","first-page":"303","volume-title":"Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics","author":"Vaught","year":"1961"},{"key":"S0022481200082232_ref001","first-page":"545","article-title":"On the categoricity in power < \u21350","volume":"7","author":"Ryll-Nardzewski","year":"1959","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences. S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200082232","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T15:53:26Z","timestamp":1559318006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200082232\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1971,9]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1971,9]]}},"alternative-id":["S0022481200082232"],"URL":"https:\/\/doi.org\/10.2307\/2269951","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1971,9]]}}}