{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T23:50:52Z","timestamp":1649116252253},"reference-count":6,"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":11972,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1981,6]]},"abstract":"<jats:p>In this paper we investigate the relationship between the number of countable and decidable models of a complete theory. The number of decidable models will be determined in two ways, in \u00a71 with respect to abstract isomorphism type, and in \u00a72 with respect to recursive isomorphism type.<\/jats:p><jats:p>Definition 1. A complete theory is (<jats:italic>\u03b1<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic>) if the number of countable models of <jats:italic>T<\/jats:italic>, up to abstract isomorphism, is <jats:italic>\u03b2<\/jats:italic>, and similarly the number of decidable models of T is <jats:italic>\u03b1<\/jats:italic>.<\/jats:p><jats:p>Definition 2. A model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200045850_inline1\" \/> is <jats:italic>\u03c9<\/jats:italic>-decidable if \u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200045850_inline1\" \/>\u2223= <jats:italic>\u03c9<\/jats:italic> and for an effective listing {<jats:italic>\u03b8<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>\u2223<jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>} of all sentences in the language of Th(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200045850_inline1\" \/>) augmented by new constant symbols <jats:italic>i<\/jats:italic>*, <jats:italic>i<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>, {<jats:italic>n<\/jats:italic> \u2223\u3008<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200045850_inline1\" \/>, <jats:italic>i<\/jats:italic>\u3009<jats:sub><jats:italic>i<\/jats:italic>&lt;<jats:italic>\u03c9<\/jats:italic><\/jats:sub> \u22a8 <jats:italic>\u03b8<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>} is recursive, where <jats:italic>i<\/jats:italic> interprets <jats:italic>i<\/jats:italic>* (in these terms, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200045850_inline1\" \/> is decidable if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200045850_inline1\" \/> is abstractly isomorphic to an <jats:italic>\u03c9<\/jats:italic>-decidable model).<\/jats:p><jats:p>Definition 3. A complete theory is (<jats:italic>\u03b1<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic>)<jats:sup><jats:italic>r<\/jats:italic><\/jats:sup> if it is (<jats:italic>\u03b3<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic>) for some <jats:italic>\u03b3<\/jats:italic> and it has exactly <jats:italic>\u03b1<\/jats:italic><jats:italic>\u03c9<\/jats:italic>-decidable models up to recursive isomorphism.<\/jats:p><jats:p>Specifically we will show in \u00a71 that there is a (2, <jats:italic>\u03c9<\/jats:italic>) theory, and in \u00a72 that although there is a (2, 2<jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup>) theory, there is no (2, <jats:italic>\u03b2<\/jats:italic>)<jats:sup><jats:italic>r<\/jats:italic><\/jats:sup> theory for any <jats:italic>\u03b2<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic> &lt; 2<jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup>.<\/jats:p>","DOI":"10.2307\/2273632","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:55:50Z","timestamp":1146952550000},"page":"397-411","source":"Crossref","is-referenced-by-count":6,"title":["Vaught's theorem recursively revisited"],"prefix":"10.1017","volume":"46","author":[{"given":"Terrence","family":"Millar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200045850_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01463352"},{"key":"S0022481200045850_ref003","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200045850_ref004","first-page":"45","volume-title":"Annals of Mathematical Logic","author":"Millar","year":"1978"},{"key":"S0022481200045850_ref001","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200045850_ref005","first-page":"307","volume":"44","author":"Millar","year":"1979","journal-title":"A complete, decidable theory with exactly two decidable models"},{"key":"S0022481200045850_ref002","volume-title":"Saturated model theory","author":"Sacks","year":"1972"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200045850","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,25]],"date-time":"2019-05-25T20:09:08Z","timestamp":1558814948000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200045850\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1981,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1981,6]]}},"alternative-id":["S0022481200045850"],"URL":"https:\/\/doi.org\/10.2307\/2273632","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1981,6]]}}}