{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T00:10:32Z","timestamp":1648512632050},"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":11150,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for <jats:italic>PC<\/jats:italic> classes.<\/jats:p><jats:p>Let <jats:italic>M<\/jats:italic> \u22a8 <jats:italic>Q<jats:sub>n<\/jats:sub>x<\/jats:italic><jats:sub>1<\/jats:sub> \u2026 <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic><jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub> \u2026 <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>) mean that there is an uncountable subset <jats:italic>A<\/jats:italic> of \u2223<jats:italic>M<\/jats:italic>\u2223 such that for every <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub> \u2026, <jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic> \u2208 <jats:italic>A<\/jats:italic>, <jats:italic>M<\/jats:italic> \u22a8 <jats:italic>\u03c6<\/jats:italic>[<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>].<\/jats:p><jats:p>Theorem 1.1 (Shelah) (\u2662<jats:sub>\u21351<\/jats:sub>). <jats:italic>For every n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic><jats:italic>the class<\/jats:italic><jats:italic>K<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub> = {\u2039<jats:italic>A, R<\/jats:italic>\u203a \u2223 \u2039<jats:italic>A, R<\/jats:italic>\u203a \u22a8 \u00ac <jats:italic>Q<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic>+1<\/jats:sup><jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub> \u2026 <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub><jats:italic>R<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub>)} <jats:italic>is not an<\/jats:italic> \u2135<jats:sub>0<\/jats:sub>-<jats:italic>PC-class in the logic<\/jats:italic> \u2112<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>, <jats:italic>obtained by closing first order logic under<\/jats:italic><jats:italic>Q<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic>. <jats:italic>I.e. for no countable<\/jats:italic> \u2112<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>-<jats:italic>theory T, is<\/jats:italic><jats:italic>K<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub><jats:italic>the class of reducts of the models of T<\/jats:italic>.<\/jats:p><jats:p>Theorem 1.2 (Rubin) (\u2662<jats:sub>\u21351<\/jats:sub>). <jats:italic>Let M<\/jats:italic> \u22a8 <jats:italic>Q<jats:sup>E<\/jats:sup> x y<\/jats:italic><jats:italic>\u03c6<\/jats:italic>(<jats:italic>x, y<\/jats:italic>) <jats:italic>mean that there is A<\/jats:italic> \u2286 \u2223<jats:italic>M<\/jats:italic>\u2223 <jats:italic>such that<\/jats:italic><jats:italic>E<\/jats:italic><jats:sub><jats:italic>A<\/jats:italic>, <jats:italic>\u03c6<\/jats:italic><\/jats:sub> = {\u2039<jats:italic>a, b<\/jats:italic>\u203a \u2223 <jats:italic>a, b<\/jats:italic> \u2208 <jats:italic>A and M<\/jats:italic> \u22a8 <jats:italic>\u03c6<\/jats:italic>[<jats:italic>a, b<\/jats:italic>]) <jats:italic>is an equivalence relation on A with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let K<jats:sup>E<\/jats:sup> = {\u2039<jats:italic>A, R<\/jats:italic>\u203a \u2223 \u2039A, R\u203a \u22a8 \u00ac Q<jats:sup>E<\/jats:sup>xyR(x, y)}. Then K<jats:sup>E<\/jats:sup> is not an \u2135<jats:sub>0<\/jats:sub>-PC-class in the logic gotten by closing first order logic under the set of quantifiers {Q<jats:sup>n<\/jats:sup> \u2223 n \u2208 <jats:italic>\u03c9<\/jats:italic>) which were defined in Theorem 1.1<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273445","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:04:45Z","timestamp":1146938685000},"page":"542-557","source":"Crossref","is-referenced-by-count":6,"title":["On the expressibility hierarchy of Magidor-Malitz quantifiers"],"prefix":"10.1017","volume":"48","author":[{"given":"Matatyahu","family":"Rubin","sequence":"first","affiliation":[]},{"given":"Saharon","family":"Shelah","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200037701_ref006","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.40.139"},{"key":"S0022481200037701_ref010","unstructured":"Rubin M. and Shelah S. , Combinatorial problems on trees. Partitions, \u22bf-systems and large free subsets, Annals of Mathematical Logic (to appear)."},{"key":"S0022481200037701_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-50-2-123-128"},{"key":"S0022481200037701_ref008","first-page":"219","volume-title":"North-Holland Studies in Logic","volume":"99","author":"Malitz","year":"1980"},{"key":"S0022481200037701_ref005","unstructured":"Kunen K. and Tall F."},{"key":"S0022481200037701_ref001","first-page":"219","article-title":"Proof of a conjecture of P. Erdos","volume":"14","author":"Fodor","year":"1952","journal-title":"Acta Universitatis Szegediensis, Acta Scientiarum Mathematicarum"},{"key":"S0022481200037701_ref011","first-page":"1","article-title":"Models with second order properties. III, Omitting types in \u03bb+ for L(Q)","volume":"21","author":"Shelah","year":"1980","journal-title":"Proceedings of the \u201cBerlin Workshop in Logic July 1977\u201d, Archiv fur Mathematische Logik und Grundlagenforschung"},{"key":"S0022481200037701_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(70)80005-5"},{"key":"S0022481200037701_ref002","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093888414"},{"key":"S0022481200037701_ref009","first-page":"439","article-title":"Arithmetical extensions with prescribed cardinality","volume":"21","author":"Rabin","year":"1959","journal-title":"Koninklijke Nederlandse Akademie von Wetenschappen, Proceedings, Series A, Mathematical Sciences"},{"key":"S0022481200037701_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90019-5"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200037701","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T18:36:33Z","timestamp":1558636593000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200037701\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1983,9]]}},"alternative-id":["S0022481200037701"],"URL":"https:\/\/doi.org\/10.2307\/2273445","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,9]]}}}