{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,29]],"date-time":"2024-06-29T06:30:21Z","timestamp":1719642621172},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12795,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,3]]},"abstract":"<jats:p>Originally generalized quantifiers were introduced to specify that a given formula was true for \u201cmany x's\u201d e.g. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/> \u22a8 Q<jats:italic>x<\/jats:italic>\u03c6(<jats:italic>x<\/jats:italic>) iff card{<jats:italic>x<\/jats:italic> \u2208 \u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/>\u2223 \u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/> \u22a8 \u03c6[<jats:italic>x<\/jats:italic>]} \u2265 \u2135<jats:sub>0<\/jats:sub>, \u2135<jats:sub>1<\/jats:sub>, or some fixed cardinal \u03ba. In this paper we formalize the notion that \u03c6{<jats:italic>x<\/jats:italic>) is true \u201cfor almost all <jats:italic>x<\/jats:italic>\u201d. This is accomplished by referring to structures <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/> = (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/>\u2032, \u03bc<jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/><\/jats:sup>) where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/>\u2032 is a first-order structure and \u03bc<jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/><\/jats:sup> is a measure of a suitable type on the universe of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048763_inline10\" \/>\u2032. We will prove that the language <jats:italic>L<\/jats:italic><jats:sub>\u03bc<\/jats:sub> obtained from first-order logic by adjoining a quantifier <jats:italic>Q<\/jats:italic><jats:sub>\u03bc<\/jats:sub>, which ranges over the measure \u03bc, is fully compact if we assume the existence of a proper class of measurable cardinals. As a corollary to the compactness theorem we obtain the recursive enumerability of the validities of <jats:italic>L<\/jats:italic><jats:sub>\u03bc<\/jats:sub>. Finally, the Magidor-Malitz quantifiers Q<jats:sub arrange=\"stack\"><jats:italic>k<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sup> (<jats:italic>n<\/jats:italic> \u2208 \u03c9) will be added to <jats:italic>L<\/jats:italic><jats:sub>\u03bc<\/jats:sub> together with analogous quantifiers Q<jats:sub arrange=\"stack\">\u03bc<\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>m<\/jats:italic><\/jats:sup> (<jats:italic>m<\/jats:italic> \u2208 \u03c9) to form <jats:italic>L<\/jats:italic><jats:sub arrange=\"stack\">\u03ba\u03bc<\/jats:sub><jats:sup arrange=\"stack\">&lt;\u03c9,&lt;\u03c9,<\/jats:sup> which is compact for sets of sentences of cardinality &lt; \u03ba, where \u03ba is a measurable cardinal &gt; \u2135<jats:sub>0<\/jats:sub>.<\/jats:p><jats:p>An alternate approach to formalizing \u201cfor almost all\u201d has been recently developed by Barwise, Kaufmann and Makkai [1] who follow a suggestion of Shelah [5].<\/jats:p>","DOI":"10.2307\/2273708","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:49:34Z","timestamp":1146952174000},"page":"103-108","source":"Crossref","is-referenced-by-count":9,"title":["The measure quantifier"],"prefix":"10.1017","volume":"44","author":[{"given":"Carl F.","family":"Morgenstern","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048763_ref006","unstructured":"Silver J. , Some applications of model theory in set theory, Ph.D. thesis, University of California, Berkeley, 1966; Annals of Mathematical Logic , vol. 3 (1971), pp. 45-110."},{"key":"S0022481200048763_ref003","unstructured":"Helling M. , Model theoretic problems for some extensions of first order languages, Ph.D. thesis, University of California, Berkeley, 1966."},{"key":"S0022481200048763_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90019-5"},{"key":"S0022481200048763_ref007","unstructured":"Slomson A.B. , Some problems in mathematical logic, Ph.D. thesis, Oxford, 1967."},{"key":"S0022481200048763_ref002","volume-title":"Set theory: An introduction to large cardinals","author":"Drake","year":"1974"},{"key":"S0022481200048763_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0376334-6"},{"key":"S0022481200048763_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90003-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048763","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T21:56:54Z","timestamp":1558907814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048763\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048763"],"URL":"https:\/\/doi.org\/10.2307\/2273708","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}