{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T13:28:53Z","timestamp":1770557333562,"version":"3.49.0"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10328,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,12]]},"abstract":"<jats:p><jats:bold>\u00a7I<\/jats:bold>. In 1961, R. L. Vaught ([V]) asked if one could prove, without the continuum hypothesis, that there exists a countable complete theory with exactly \u2135<jats:sub>1<\/jats:sub> isomorphism types of countable models. The following statement is known as Vaught conjecture:<\/jats:p><jats:p><jats:italic>Let T be a countable theory. If T has uncountably many countable models, then T has<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031959_inline1\"\/><jats:italic>countable models<\/jats:italic>.<\/jats:p><jats:p>More than twenty years later, this question is still open. Many papers have been written on the question: see for example [HM], [M1], [M2] and [St]. In the opinion of many people, it is a major problem in model theory.<\/jats:p><jats:p>Of course, I cannot say what Vaught had in mind when he asked the question. I just want to explain here what meaning I personally see to this problem. In particular, I will not speak about the topological Vaught conjecture, which is quite another issue.<\/jats:p><jats:p>I suppose that the first question I shall have to face is the following: \u201cWhy on earth are you interested in the number of countable models\u2014particularly since the whole question disappears if we assume the continuum hypothesis?\u201d The answer is simply that I am not interested in the number of countable models, nor in the number of models in any cardinality, as a matter of fact. An explanation is due here; it will be a little technical and it will rest upon two names: Scott (sentences) and Morley (theorem).<\/jats:p>","DOI":"10.2307\/2273984","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:15:32Z","timestamp":1146953732000},"page":"973-982","source":"Crossref","is-referenced-by-count":5,"title":["Why some people are excited by Vaught's conjecture"],"prefix":"10.1017","volume":"50","author":[{"given":"Daniel","family":"Lascar","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031959_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70961-3"},{"key":"S0022481200031959_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069301"},{"key":"S0022481200031959_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1977-0472506-8"},{"key":"S0022481200031959_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757141"},{"key":"S0022481200031959_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761237"},{"key":"S0022481200031959_ref015","unstructured":"Shelah S. , New edition of [Sh1] (to appear)."},{"key":"S0022481200031959_ref005","first-page":"45","volume-title":"Publication des Groupes de Contact","author":"Lascar","year":"1981"},{"key":"S0022481200031959_ref001","first-page":"5","volume-title":"Studies in model theory","author":"Barwise","year":"1973"},{"key":"S0022481200031959_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760643"},{"key":"S0022481200031959_ref004","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF02760651","article-title":"A proof of Vaught's conjecture for \u03c9-stable theories","volume":"49","author":"Harrington","year":"1984","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200031959_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90008-0"},{"key":"S0022481200031959_ref007","first-page":"301","volume":"46","author":"Makkai","year":"1981","journal-title":"An example concerning Scott height"},{"key":"S0022481200031959_ref008","first-page":"14","volume":"35","author":"Morley","year":"1970","journal-title":"The number of countable models"},{"key":"S0022481200031959_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90017-5"},{"key":"S0022481200031959_ref012","first-page":"329","volume-title":"The theory of models","author":"Scott","year":"1965"},{"key":"S0022481200031959_ref013","volume-title":"Classification theory and the number of nonisomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200031959_ref017","first-page":"303","volume-title":"Infinitistic methods (Proceedings of the symposium on the foundations of mathematics, Warsaw, 1959)","author":"Vaught","year":"1961"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031959","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T20:56:02Z","timestamp":1558558562000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031959\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,12]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1985,12]]}},"alternative-id":["S0022481200031959"],"URL":"https:\/\/doi.org\/10.2307\/2273984","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,12]]}}}