{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T09:29:39Z","timestamp":1772443779996,"version":"3.50.1"},"reference-count":8,"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":14529,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,6]]},"abstract":"<jats:p>It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable \u2135<jats:sub>1<\/jats:sub>-categorical theory.<\/jats:p><jats:p>The language of a theory <jats:italic>T<\/jats:italic> is denoted by <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>). All languages will be presumed countable. An <jats:italic>x-type<\/jats:italic> of <jats:italic>T<\/jats:italic> is a set of formulas with free variables <jats:italic>x<\/jats:italic>, which is consistent with <jats:italic>T<\/jats:italic> and which is maximal in this property. A formula with free variables <jats:italic>x<\/jats:italic> is <jats:italic>complete<\/jats:italic> if there is exactly one <jats:italic>x<\/jats:italic>-type containing it. A type is <jats:italic>principal<\/jats:italic> if it contains a complete formula. A countable model of <jats:italic>T<\/jats:italic> is <jats:italic>prime<\/jats:italic> if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.<\/jats:p><jats:p>If <jats:italic>T<\/jats:italic> is a decidable theory, then the decision procedure for <jats:italic>T<\/jats:italic> equips <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>) with an effective counting. Thus the formulas of <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>) correspond to integers. The integer a formula <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>) corresponds to is generally called the G\u00f6del number of <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>) and is denoted by \u231c<jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>)\u231d. The usual recursion theoretic notions defined on the set of integers can be transferred to <jats:italic>L<\/jats:italic>(<jats:italic>T<\/jats:italic>). In particular a type \u0393 is <jats:italic>recursive<\/jats:italic> with index <jats:italic>e<\/jats:italic> if {\u231c<jats:italic>\u03c6<\/jats:italic>\u231d.; <jats:italic>\u03c6<\/jats:italic> \u2208 \u0393} is a recursive set of integers with index <jats:italic>e<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2272643","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:29:41Z","timestamp":1146936581000},"page":"305-309","source":"Crossref","is-referenced-by-count":52,"title":["Recursively presentable prime models"],"prefix":"10.1017","volume":"39","author":[{"given":"Leo","family":"Harrington","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200064471_ref004","volume-title":"An introduction to differential algebra","author":"Kaplansky","year":"1957"},{"key":"S0022481200064471_ref002","first-page":"79","volume":"36","author":"Baldwin","year":"1971","journal-title":"On strongly minimal sets"},{"key":"S0022481200064471_ref003","volume-title":"Proceedings of the 3rd International Congress for Logic, Methodology and Philosophy of Science","author":"Ershov","year":"1967"},{"key":"S0022481200064471_ref007","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200064471_ref008","unstructured":"Wood C. , Prime model extensions for differential fields of characteristic p \u2260 0, this Journal (to appear)."},{"key":"S0022481200064471_ref006","volume-title":"Recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200064471_ref005","first-page":"341","article-title":"Computable algebra","volume":"95","author":"Rabin","year":"1960","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200064471_ref001","first-page":"37","article-title":"\u03b1\u03c4 is finite for \u21351-categorical T","volume":"181","author":"Baldwin","year":"1973","journal-title":"Transactions of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200064471","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:19:53Z","timestamp":1559150393000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200064471\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1974,6]]}},"alternative-id":["S0022481200064471"],"URL":"https:\/\/doi.org\/10.2307\/2272643","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,6]]}}}