{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T04:20:45Z","timestamp":1649046045006},"reference-count":13,"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":11515,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,9]]},"abstract":"<jats:p>One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence.<\/jats:p><jats:p>In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including <jats:italic>L(Q)<\/jats:italic> and <jats:italic>L(aa)<\/jats:italic>, respectively the languages with the quantifiers \u201cthere exist un-countably many\u201d and \u201cfor almost all countable subsets\u201d.<\/jats:p><jats:p>In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for <jats:italic>L(Q)<\/jats:italic>. We also consider whether the natural class of \u201cgeneralized \u03a3-sentences\u201d in <jats:italic>L(Q)<\/jats:italic> characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open.<\/jats:p><jats:p>In the third section we show that the analogous class of <jats:italic>L(aa)<\/jats:italic>-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3].<\/jats:p><jats:p>In the fourth section we show that under rather general conditions, if <jats:italic>L<\/jats:italic> is a countably compact extension of first-order logic and <jats:italic>T<\/jats:italic> is an \u2135<jats:sub>1<\/jats:sub>-categorical first-order theory, then <jats:italic>L<\/jats:italic> is trivial relative to <jats:italic>T<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273589","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:00:58Z","timestamp":1146952858000},"page":"572-586","source":"Crossref","is-referenced-by-count":0,"title":["Some contributions to definability theory for languages with generalized quantifiers"],"prefix":"10.1017","volume":"47","author":[{"given":"John T.","family":"Baldwin","sequence":"first","affiliation":[]},{"given":"Douglas E.","family":"Miller","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043966_ref003","first-page":"304","volume":"43","author":"Bruce","year":"1978","journal-title":"Ideal models and some not so ideal problems in the model theory of L(Q)"},{"key":"S0022481200043966_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90003-7"},{"key":"S0022481200043966_ref001","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1980.90.11"},{"key":"S0022481200043966_ref010","first-page":"33","volume":"21","author":"Robinson","year":"1956","journal-title":"Note on a problem of L. Henkin"},{"key":"S0022481200043966_ref013","doi-asserted-by":"publisher","DOI":"10.4064\/fm-54-3-303-304"},{"key":"S0022481200043966_ref011","first-page":"249","volume-title":"Set Theory and Hierarchy Theory, Lecture Notes in Mathematics","volume":"537","author":"Slomson","year":"1976"},{"key":"S0022481200043966_ref004","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200043966_ref006","first-page":"121","volume-title":"Theory of models","author":"Fuhrken","year":"1965"},{"key":"S0022481200043966_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90009-3"},{"key":"S0022481200043966_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(70)80005-5"},{"key":"S0022481200043966_ref009","first-page":"115","article-title":"Some model theory for monotone quantifiers","volume":"13","author":"Makowsky","year":"1975","journal-title":"Archir f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"S0022481200043966_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90019-5"},{"key":"S0022481200043966_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(54)50074-0"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043966","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T20:47:15Z","timestamp":1558730835000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043966\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1982,9]]}},"alternative-id":["S0022481200043966"],"URL":"https:\/\/doi.org\/10.2307\/2273589","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,9]]}}}