{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T15:23:55Z","timestamp":1775834635944,"version":"3.50.1"},"reference-count":9,"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":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>Recursive model theory is supposed to be the study of the effectiveness of constructions and theorems in model theory. This often involves getting \u201ceffective\u201d versions of various classical model-theoretic notions. The traditional way of doing this is to restrict attention to recursive models, and recursive isomorphisms between them, etc. Thus for example the following definition appears in the literature (in [3] and [1]).<\/jats:p><jats:p>Definition. Given a recursive model <jats:italic>A<\/jats:italic> and an <jats:italic>n<\/jats:italic> \u0404 \u03c9, a subset <jats:italic>R<\/jats:italic> \u2286 <jats:italic>A<jats:sup>n<\/jats:sup><\/jats:italic> is called <jats:italic>intrinsically r.e.<\/jats:italic> provided that for every recursive model <jats:italic>B<\/jats:italic> \u2248 <jats:italic>A<\/jats:italic>, the isomorphic image in <jats:italic>B<\/jats:italic> of <jats:italic>R<\/jats:italic> is an r.e. subset of <jats:italic>B<jats:sup>n<\/jats:sup><\/jats:italic>.<\/jats:p><jats:p>It is clear that if <jats:italic>R<\/jats:italic> is definable by a (recursive, infinitary) \u03a3<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup> formula (with finitely many parameters from <jats:italic>A<\/jats:italic>), then <jats:italic>R<\/jats:italic> is intrinsically r.e. It seems natural for the converse to be true. Indeed, provided that (<jats:italic>A, R<\/jats:italic>) is sufficiently \u201cregular\u201d in a sense made precise in a theorem of Ash and Nerode (see [3]), the converse is true. However, if we drop the (rather strong) regularity conditions, there exist \u201cpathological\u201d examples of intrinsically r.e. relations which are not definable by a \u03a3<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup> formula (see [7]).<\/jats:p><jats:p>In this paper, we suggest a rather different approach to studying the effectiveness of model theory, an approach we have dubbed \u201ceffective model theory\u201d. The basic idea is to allow arbitrary nonrecursive models, but to require all notions to be relativized to the complexity of the models involved. (Much the same notion has been used in [2] under the name \u201crelatively recursive model theory\u201d.) Thus for example we have the following effective model theory version of the property of being intrinsically r.e.<\/jats:p>","DOI":"10.2307\/2274481","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:36:50Z","timestamp":1146955010000},"page":"1168-1191","source":"Crossref","is-referenced-by-count":77,"title":["Effective model theory vs. recursive model theory"],"prefix":"10.1017","volume":"55","author":[{"given":"John","family":"Chisholm","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025639_ref009","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200025639_ref008","unstructured":"Manasse Mark and Slaman Theodore A. , Immutably r.e. relations (unpublished)."},{"key":"S0022481200025639_ref006","first-page":"1034","volume":"51","author":"Knight","year":"1986","journal-title":"Degrees coded in jumps of orderings"},{"key":"S0022481200025639_ref005","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200025639_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01669456"},{"key":"S0022481200025639_ref003","first-page":"26","volume-title":"Aspects of recursive algebra","author":"Ash","year":"1981"},{"key":"S0022481200025639_ref002","unstructured":"Ash C. J. et al., Generic copies of countable structures (unpublished)."},{"key":"S0022481200025639_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90048-5"},{"key":"S0022481200025639_ref007","unstructured":"Manasse Mark Steven , Techniques and counterexamples in almost categorical recursive model theory, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin, 1982."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025639","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:25:34Z","timestamp":1558211134000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025639\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025639"],"URL":"https:\/\/doi.org\/10.2307\/2274481","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}