{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:52:41Z","timestamp":1775461961022,"version":"3.50.1"},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12795,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,3]]},"abstract":"<jats:p>A structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048726_inline10\"\/> is <jats:italic>recursive<\/jats:italic> if the set of quantifier-free sentences in the complete diagram \u22bf(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048726_inline10\"\/>) of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048726_inline10\"\/> is recursive. It has been known for some time that every decidable theory has a recursive model. In fact, every decidable theory has a decidable model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048726_inline10\"\/> (that is a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048726_inline10\"\/> such that \u22bf(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048726_inline10\"\/>) is recursive). In this paper we find other conditions which imply that a theory have a recursive model.<\/jats:p><jats:p>In \u00a71 we study the relation between an \u2135<jats:sub>0<\/jats:sub>-categorical theory <jats:italic>T<\/jats:italic> having a recursive model and the complexity of the quantificational hierarchy of that theory. We let \u2203<jats:sub>0<\/jats:sub> denote the set of quantifier-free sentences, and let \u2203<jats:sub><jats:italic>n<\/jats:italic>\u00f71<\/jats:sub> denote the set of sentences beginning with an existential quantifier and having <jats:italic>n<\/jats:italic> alternations of quantifiers. (\u2200<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is defined analogously.) Then we show that if <jats:italic>T<\/jats:italic> is an arithmetical \u2135<jats:sub>0<\/jats:sub>-categorical theory such that <jats:italic>T<\/jats:italic> \u22c2 \u2203<jats:sub><jats:italic>n<\/jats:italic>\u00f72<\/jats:sub> is \u03a3<jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic>\u00f71<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup> for each <jats:italic>n<\/jats:italic> &lt; \u03c9, then <jats:italic>T<\/jats:italic> has a recursive model. We show that this is a best possible result by giving an example of a \u22bf<jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic>\u00f72<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup> \u2135<jats:sub>0<\/jats:sub>-categorical theory <jats:italic>T<\/jats:italic> such that <jats:italic>T<\/jats:italic> \u22c2 \u2203<jats:sub><jats:italic>n<\/jats:italic>\u00f71<\/jats:sub> is recursive yet <jats:italic>T<\/jats:italic> has no recursive model.<\/jats:p><jats:p>In \u00a72 we consider the theory of trees. Ershov [1] had proved that every \u03a3<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup> theory of trees has a recursive model. We show this to be best possible by giving an example of a \u22bf<jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup> theory of trees which has no recursive model.<\/jats:p>","DOI":"10.2307\/2273704","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:49:34Z","timestamp":1146937774000},"page":"59-76","source":"Crossref","is-referenced-by-count":20,"title":["Theories with recursive models"],"prefix":"10.1017","volume":"44","author":[{"given":"Manuel","family":"Lerman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James H.","family":"Schmerl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048726_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02218728"},{"key":"S0022481200048726_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219295"},{"key":"S0022481200048726_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219289"},{"key":"S0022481200048726_ref004","first-page":"494","volume":"37","author":"Henson","year":"1972","journal-title":"Countable homogeneous relational structures and \u21350-categorical theories"},{"key":"S0022481200048726_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm-59-1-109-116"},{"key":"S0022481200048726_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90030-X"},{"key":"S0022481200048726_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757002"},{"key":"S0022481200048726_ref003","first-page":"305","volume":"39","author":"Harrington","year":"1974","journal-title":"Recursively presentable prime models"},{"key":"S0022481200048726_ref008","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100003844"},{"key":"S0022481200048726_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02218589"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048726","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T17:57:04Z","timestamp":1558893424000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048726\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048726"],"URL":"https:\/\/doi.org\/10.2307\/2273704","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}