{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T00:46:10Z","timestamp":1649033170325},"reference-count":1,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7041,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,12]]},"abstract":"The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].<\/jats:p>Let L<\/jats:italic> be a countable first-order language. Let L<\/jats:italic>\u2032 be the relational language having, for each formula \u03c6<\/jats:italic> of L<\/jats:italic> and each sequence \u03c5<\/jats:italic>1<\/jats:sub>,\u2026,\u03c5<\/jats:italic>n<\/jats:italic><\/jats:sub> of variables including the free variables of \u03c6<\/jats:italic>, an n<\/jats:italic>-ary relation symbol P<\/jats:italic>\u03c6<\/jats:italic><\/jats:sub>. For any L<\/jats:italic>-structure and any formula \u03a8<\/jats:italic>(\u03c5<\/jats:italic>) of L<\/jats:italic>, we define the \u03a8-fraction of<\/jats:italic> to be the L<\/jats:italic>\u2032-structure \u03a8<\/jats:italic><\/jats:sub> whose universe consists of those elements of satisfying \u03a8<\/jats:italic>(\u03c5<\/jats:italic>) and whose relations {R<\/jats:italic>\u03c6<\/jats:italic><\/jats:sub>}\u03c6\u03f5L<\/jats:italic><\/jats:sub> are defined by letting a<\/jats:italic>1<\/jats:sub>,\u2026,a<\/jats:italic>n<\/jats:italic><\/jats:sub> satisfy R\u03c6<\/jats:sub><\/jats:italic> in \u03a8<\/jats:italic><\/jats:sub> if, and only if, a<\/jats:italic>1<\/jats:sub>,\u2026, a<\/jats:italic>n<\/jats:italic><\/jats:sub> satisfy \u03c6<\/jats:italic> in .<\/jats:p>An L-elementary class<\/jats:italic> means the class of all L<\/jats:italic>-structures satisfying each of some set of sentences of L<\/jats:italic>. The countable part<\/jats:italic> of an L<\/jats:italic>-elementary class K<\/jats:italic> means the class of all countable L<\/jats:italic>-structures from K<\/jats:italic>.<\/jats:p>Theorem. Let K be an L-elementary class and let \u03a8<\/jats:italic>(\u03c5<\/jats:italic>) be a formula of L. Then the class of countable \u03a8<\/jats:italic>-fractions of structures in K is the countable part of some L\u2032-elementary class.<\/jats:italic><\/jats:p>Comment<\/jats:italic>. By the downward L\u00f6wenheim-Skolem theorem, the countable \u03a8<\/jats:italic>-fractions of structures in K<\/jats:italic> are the same as the \u03a8<\/jats:italic>-fractions of countable structures in K<\/jats:italic>.<\/jats:p>Proof. We give a set \u03a3\u2032 of L<\/jats:italic>\u2032-sentences whose countable models are exactly the countable \u03a8<\/jats:italic>-fractions of structures in K<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275713","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:53:29Z","timestamp":1146956009000},"page":"1410-1413","source":"Crossref","is-referenced-by-count":0,"title":["On countable fractions from an elementary class"],"prefix":"10.1017","volume":"59","author":[{"given":"C. J.","family":"Ash","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019356_ref001","volume-title":"Model theory","author":"Chang","year":"1973"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019356","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T00:41:23Z","timestamp":1557880883000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019356\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":1,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["S0022481200019356"],"URL":"http:\/\/dx.doi.org\/10.2307\/2275713","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":["Logic","Philosophy"],"published":{"date-parts":[[1994,12]]}}}