{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,16]],"date-time":"2023-10-16T06:59:33Z","timestamp":1697439573224},"reference-count":16,"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":9781,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We associate with any abstract logic <jats:italic>L<\/jats:italic> a family <jats:bold>F<\/jats:bold>(<jats:italic>L<\/jats:italic>) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of <jats:italic>L<\/jats:italic>.<\/jats:p><jats:p>For every countably generated [<jats:italic>\u03c9, \u03c9<\/jats:italic>]-compact logic <jats:italic>L<\/jats:italic>, our main applications are:<\/jats:p><jats:p>(i) Elementary classes of <jats:italic>L<\/jats:italic> can be characterized in terms of \u2261<jats:sub><jats:italic>L<\/jats:italic><\/jats:sub> only.<\/jats:p><jats:p>(ii) If <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200034393_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200034393_inline2\" \/> are countable models of a countable superstable theory without the finite cover property, then <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200034393_inline3\" \/>.<\/jats:p><jats:p>(iii) There exists the \u201clargest\u201d logic <jats:italic>M<\/jats:italic> such that complete extensions in the sense of <jats:italic>M<\/jats:italic> and <jats:italic>L<\/jats:italic> are the same; moreover <jats:italic>M<\/jats:italic> is still [\u03c9, \u03c9]-compact and satisfies an interpolation property stronger than unrelativized \u22bf-closure.<\/jats:p><jats:p>(iv) If <jats:italic>L<\/jats:italic> = <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9\u03c9<\/jats:italic><\/jats:sub>(<jats:italic>Q<jats:sub>x<\/jats:sub><\/jats:italic>), then cf(<jats:italic>\u03c9<\/jats:italic><jats:sub><jats:italic>x<\/jats:italic><\/jats:sub>) &gt; <jats:italic>\u03c9<\/jats:italic> and <jats:italic>\u03bb<jats:sup>\u03c9<\/jats:sup> &lt; \u03c9<\/jats:italic><jats:sub><jats:italic>x<\/jats:italic><\/jats:sub>, for all <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic><jats:sub><jats:italic>x<\/jats:italic><\/jats:sub>.<\/jats:p><jats:p>We also prove that no proper extension of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9\u03c9<\/jats:italic><\/jats:sub> generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03ba\u03bb<\/jats:italic><\/jats:sub>-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.<\/jats:p>","DOI":"10.2307\/2274393","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:21:40Z","timestamp":1146954100000},"page":"437-454","source":"Crossref","is-referenced-by-count":6,"title":["Limit ultrapowers and abstract logics"],"prefix":"10.1017","volume":"52","author":[{"given":"Paolo","family":"Lipparini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034393_ref016","volume-title":"Classification theory and the number of nonisomorphic models","volume":"92","author":"Shelah","year":"1978"},{"key":"S0022481200034393_ref010","first-page":"569","article-title":"Robinson equivalence relations through limit ultrapowers","volume":"4","author":"Lipparini","year":"1985","journal-title":"Bollettino dell'Unione Matematica Italiana"},{"key":"S0022481200034393_ref015","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1982-0642332-1"},{"key":"S0022481200034393_ref011","first-page":"42","volume":"50","author":"Makowsky","year":"1985","journal-title":"Vop\u011bnka's principle and compact logics"},{"key":"S0022481200034393_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF02318027"},{"key":"S0022481200034393_ref004","doi-asserted-by":"crossref","unstructured":"Ebbinghaus H. D. , Extended logics: the general framework, in [BF], pp. 25\u201376.","DOI":"10.1017\/9781316717158.005"},{"key":"S0022481200034393_ref003","volume-title":"Model theory","volume":"73","author":"Chang","year":"1977"},{"key":"S0022481200034393_ref006","volume-title":"Model theory and its applications","author":"Kopperman","year":"1972"},{"key":"S0022481200034393_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90021-0"},{"key":"S0022481200034393_ref002","first-page":"1","volume-title":"Methods in mathematical logic (proceedings of the sixth Latin American symposium in mathematical logic","volume":"1130","author":"Caicedo","year":"1985"},{"key":"S0022481200034393_ref012","first-page":"215","article-title":"The theorems of Beth and Craig in abstract model theory. I: The abstract setting","volume":"256","author":"Makowsky","year":"1979","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200034393_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1964-11219-2"},{"key":"S0022481200034393_ref001","volume-title":"Model-theoretic logics","volume":"6","author":"Barwise","year":"1985"},{"key":"S0022481200034393_ref007","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1969.tb00356.x"},{"key":"S0022481200034393_ref008","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19850313107"},{"key":"S0022481200034393_ref009","first-page":"491","volume":"51","author":"Lipparini","year":"1986","journal-title":"Logics closed under first order unary operations"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034393","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T19:48:25Z","timestamp":1558468105000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/product\/identifier\/S0022481200034393\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,6]]},"references-count":16,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,6]]}},"alternative-id":["S0022481200034393"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.2307\/2274393","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,6]]}}}