{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,12]],"date-time":"2022-06-12T08:26:05Z","timestamp":1655022365291},"reference-count":11,"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":5306,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,9]]},"abstract":"<jats:p>In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if <jats:italic>M<\/jats:italic> is a model of a theory <jats:italic>T<\/jats:italic> in a language <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200013189_inline1\" \/>, then, clearly, any definable subset <jats:italic>S<\/jats:italic> \u2282 <jats:italic>M<\/jats:italic> (i.e., a subset <jats:italic>S<\/jats:italic> = {<jats:italic>a<\/jats:italic> \u2223 <jats:italic>M<\/jats:italic> \u22a8 \u03c6(<jats:italic>a<\/jats:italic>)} defined by some formula \u03c6) is invariant under all automorphisms of <jats:italic>M<\/jats:italic>. The same is of course true for subsets of <jats:italic>M<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup><\/jats:italic> defined by formulas with <jats:italic>n<\/jats:italic> free variables.<\/jats:p><jats:p>Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory <jats:italic>T<\/jats:italic> we will construct a Boolean valued model <jats:italic>M<\/jats:italic>, in which precisely the <jats:italic>T<\/jats:italic> -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of <jats:italic>M<\/jats:italic> is definable by a formula <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200013189_inline1\" \/>.<\/jats:p><jats:p>Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.<\/jats:p>","DOI":"10.2307\/2586617","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:03:11Z","timestamp":1146924191000},"page":"1028-1036","source":"Crossref","is-referenced-by-count":4,"title":["An elementary definability theorem for first order logic"],"prefix":"10.1017","volume":"64","author":[{"given":"C.","family":"Butz","sequence":"first","affiliation":[]},{"given":"I.","family":"Moerdijk","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013189_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19970430304"},{"key":"S0022481200013189_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)00042-0"},{"key":"S0022481200013189_ref003","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/S0022-4049(97)00107-2","article-title":"Representing topoi by topological groupoids","volume":"130","author":"Burz","year":"1998","journal-title":"Journal of Pure and Applied Algebra"},{"key":"S0022481200013189_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90042-9"},{"key":"S0022481200013189_ref010","first-page":"329","volume-title":"The theory of models","author":"Scott","year":"1965"},{"key":"S0022481200013189_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0927-0"},{"key":"S0022481200013189_ref011","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1959.tb00301.x"},{"key":"S0022481200013189_ref006","first-page":"19","article-title":"Booleschewertige Logik","volume":"87","author":"Koppelberg","year":"1985","journal-title":"Jber. d. dt. Math.-Verein"},{"key":"S0022481200013189_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200013189_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079688"},{"key":"S0022481200013189_ref001","first-page":"330","article-title":"On Padoa's method in the theory of definition","volume":"56","author":"Beth","year":"1953","journal-title":"Nederl Acad. Wetensch. Proc. Ser. A"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013189","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T16:37:35Z","timestamp":1557419855000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013189\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1999,9]]}},"alternative-id":["S0022481200013189"],"URL":"https:\/\/doi.org\/10.2307\/2586617","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,9]]}}}