{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:21:55Z","timestamp":1775463715983,"version":"3.50.1"},"reference-count":1,"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":10146,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,6]]},"abstract":"<jats:p>This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If <jats:italic>\u03b8<\/jats:italic> is a formula, then <jats:italic>\u03b8<jats:sup>k<\/jats:sup><\/jats:italic> denotes <jats:italic>\u03b8<\/jats:italic> if <jats:italic>k<\/jats:italic> is zero, and \u00ac<jats:italic>\u03b8<\/jats:italic> if <jats:italic>k<\/jats:italic> is one. If <jats:italic>A<\/jats:italic> is a sequence with domain a subset of <jats:italic>\u03c9<\/jats:italic>, then <jats:italic>A<\/jats:italic>\u2223<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> denotes the sequence obtained by restricting the domain of <jats:italic>A<\/jats:italic> to <jats:italic>n<\/jats:italic>. For an effective first order language <jats:italic>L<\/jats:italic>, let {<jats:italic>c<jats:sub>i<\/jats:sub><\/jats:italic>\u2223<jats:italic>i<\/jats:italic>&lt;<jats:italic>\u03c9<\/jats:italic>} be distinct new constants, and let {<jats:italic>\u03b8<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2223<jats:italic>i<\/jats:italic>&lt;<jats:italic>\u03c9<\/jats:italic>} be an effective enumeration of all sentences in the language <jats:italic>L<\/jats:italic> \u222a {<jats:italic>c<jats:sub>i<\/jats:sub><\/jats:italic>\u2223<jats:italic>j<\/jats:italic>&lt;<jats:italic>\u03c9<\/jats:italic>}. An infinite L-structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031297_inline1.png\"\/> is <jats:italic>recursive<\/jats:italic> iff it has universe <jats:italic>\u03c9<\/jats:italic> and the set <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031297_inline2.png\"\/> is recursive, where <jats:italic>c<jats:sub>n<\/jats:sub><\/jats:italic> is interpreted by <jats:italic>n<\/jats:italic>. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The \u2203-diagram of a structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031297_inline1.png\"\/> is recursive if the structure is recursive and the set <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031297_inline3.png\"\/> and <jats:italic>\u03b8<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> is an existential sentence} is also recursive. The definition of \u201cthe \u2200\u2203-diagram of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031297_inline1.png\"\/> is recursive\u201d is similar.<\/jats:p>","DOI":"10.2307\/2274066","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:17:35Z","timestamp":1146953855000},"page":"430-434","source":"Crossref","is-referenced-by-count":16,"title":["Recursive categoricity and persistence"],"prefix":"10.1017","volume":"51","author":[{"given":"Terrence","family":"Millar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031297_bib001","first-page":"647","article-title":"Self-stability and computable families of constructivizations","volume":"14","author":"Gon\u010darov","year":"1975","journal-title":"Alg\u00e9bra i Logika"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031297","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:39:25Z","timestamp":1679481565000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031297\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,6]]},"references-count":1,"aliases":["10.1017\/s0022481200031297","10.1017\/s0022481200031297"],"journal-issue":{"issue":"2","published-print":{"date-parts":[[1986,6]]}},"alternative-id":["S0022481200031297"],"URL":"https:\/\/doi.org\/10.2307\/2274066","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,6]]}}}