{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T02:26:54Z","timestamp":1667960814533},"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":12611,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,9]]},"abstract":"<jats:p>A well-known result of Vaught's is that no complete theory has exactly two nonisomorphic countable models. The main result of this paper is that there is a complete decidable theory with exactly two nonisomorphic decidable models.<\/jats:p><jats:p>A model is decidable if it has a decidable satisfaction predicate. To be more precise, let <jats:italic>T<\/jats:italic> be a decidable theory, let {\u03b8<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>\u2223<jats:italic>n<\/jats:italic> &lt; \u03c9} be an effective enumeration of all formulas in <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>), and let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048222_inline1\" \/> be a countable model of <jats:italic>T<\/jats:italic>. For any indexing <jats:italic>E<\/jats:italic> = {<jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic>\u2223 <jats:italic>i<\/jats:italic> &lt; \u03c9} of \u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048222_inline1\" \/>\u2223, and any formula \u03d5 \u2208 <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>), let \u2018\u03d5<jats:sup><jats:italic>E<\/jats:italic><\/jats:sup>\u2019 denote the result of substituting \u2018<jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic>\u2019 for every free occurrence of \u2018<jats:italic>x<jats:sub>i<\/jats:sub><\/jats:italic>\u2019 in \u03d5, <jats:italic>i<\/jats:italic> &lt; \u03c9. Then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048222_inline1\" \/> is decidable just in case, for some indexing <jats:italic>E<\/jats:italic> of \u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048222_inline1\" \/>\u2223, {<jats:italic>n<\/jats:italic> \u2223 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048222_inline1\" \/> \u22a8 \u03b8<jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>E<\/jats:italic><\/jats:sup>} is a recursive set of integers. It is easy to show that the decidability of a model does not depend on the choice of the effective enumeration of the formulas in <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>); we omit details. By a simple \u2018effectivization\u2019 of Henkin's proof of the completeness theorem (see Chang [1]) we have<\/jats:p><jats:p><jats:italic>Fact<\/jats:italic> 1. Every decidable consistent theory has a decidable model.<\/jats:p><jats:p>Assume next that <jats:italic>T<\/jats:italic> is a complete decidable theory and {\u03b8<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> \u2223 <jats:italic>n<\/jats:italic> &lt; \u03c9} is an effective enumeration of all formulas of <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2273123","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:50:56Z","timestamp":1146952256000},"page":"307-312","source":"Crossref","is-referenced-by-count":8,"title":["A complete, decidable theory with two decidable models"],"prefix":"10.1017","volume":"44","author":[{"given":"Terrence S.","family":"Millar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048222_ref002","first-page":"305","volume":"39","author":"Harrington","year":"1974","journal-title":"Recursively presentable prime models"},{"key":"S0022481200048222_ref005","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200048222_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90030-X"},{"key":"S0022481200048222_ref003","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BFb0062858","volume-title":"Algebra and logic, Lecture Notes in Mathematics","author":"Metakides","year":"1975"},{"key":"S0022481200048222_ref006","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200048222_ref001","volume-title":"Model theory","author":"Chang","year":"1973"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048222","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T20:53:01Z","timestamp":1558903981000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048222\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S0022481200048222"],"URL":"https:\/\/doi.org\/10.2307\/2273123","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}