{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T18:08:32Z","timestamp":1649095712336},"reference-count":21,"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":11242,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,6]]},"abstract":"<jats:p>The incompleteness of ZF set theory leads one to look for natural extensions of ZF in which one can prove statements independent of ZF which appear to be \u201ctrue\u201d. One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM. In this paper we look at a set theory ZF(aa), with an added quantifier aa which ranges over ordinals. The \u201caa\u201d stands for \u201calmost all\u201d, and although we will consider interpretations in terms of the closed unbounded filter on a regular cardinal \u03ba, we will consider other interpretations also.<\/jats:p><jats:p>We start in \u00a71 by giving the axioms for the theory ZF(aa) and presenting a completeness theorem which gives a model-theoretic definition of ZF(aa). In \u00a72 we investigate set theory with a satisfaction predicate and interpret it in a fragment of ZF(aa). In \u00a73 we generalize the methods of \u00a72 to obtain a hierarchy of satisfaction predicates. We use these predicates to prove reflection theorems, as well as to prove the consistency of certain fragments of ZF(aa). Next, in \u00a74 we discuss expandability of models of ZF to models of fragments of ZF(aa) and of Kelley-Morse. We conclude in \u00a75 with a discussion of an extension ZF(aa) + DET of ZF(aa) in which the quantifier aa is self-dual.<\/jats:p>","DOI":"10.2307\/2273546","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:03:20Z","timestamp":1146953000000},"page":"263-287","source":"Crossref","is-referenced-by-count":5,"title":["Set theory with a filter quantifier"],"prefix":"10.1017","volume":"48","author":[{"given":"Matt","family":"Kaufmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120003824X_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90013-4"},{"key":"S002248120003824X_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-71-1-43-62"},{"key":"S002248120003824X_ref021","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-2-219-242"},{"key":"S002248120003824X_ref002","volume-title":"Annals of Mathematical Logic","author":"Barwise"},{"key":"S002248120003824X_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0376334-6"},{"key":"S002248120003824X_ref003","first-page":"109","volume-title":"Infinite and finite sets","volume":"I","author":"Baumgartner","year":"1975"},{"key":"S002248120003824X_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90011-5"},{"key":"S002248120003824X_ref004","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S002248120003824X_ref007","unstructured":"Kaufmann M. , Some results in stationary logic, Doctoral dissertation, University of Wisconsin-Madison, 1978."},{"key":"S002248120003824X_ref013","unstructured":"Ratajczyk Z. , On axiomatization of ZFKM and ZFKM (to appear)."},{"key":"S002248120003824X_ref009","first-page":"186","volume-title":"Proceedings of a Logic Symposium (Karpacz 1979)","volume":"834","author":"Macintyre","year":"1980"},{"key":"S002248120003824X_ref010","first-page":"203","volume-title":"Logic Conference Kiel 1974, Lecture Notes in Mathematics","volume":"537","author":"Marek","year":"1976"},{"key":"S002248120003824X_ref011","first-page":"187","volume":"47","author":"Morgenstern","year":"1982","journal-title":"On generalized quantifiers in arithmetic"},{"key":"S002248120003824X_ref020","unstructured":"Kakuda Y. , notes."},{"key":"S002248120003824X_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90003-7"},{"key":"S002248120003824X_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90009-3"},{"key":"S002248120003824X_ref018","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0280359"},{"key":"S002248120003824X_ref019","volume-title":"Set theory based on the language with the additional quantifier \u201cfor almost all\u201d. I","author":"Kakuda","year":"1980"},{"key":"S002248120003824X_ref008","volume-title":"Fundamenta Mathematicae","author":"Kaufmann"},{"key":"S002248120003824X_ref016","first-page":"423","volume":"47","author":"Schmerl","year":"1982","journal-title":"On the role of Ramsey quantifiers in first-order arithmetic"},{"key":"S002248120003824X_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF02121264"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120003824X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T19:06:24Z","timestamp":1558724784000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120003824X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,6]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1983,6]]}},"alternative-id":["S002248120003824X"],"URL":"https:\/\/doi.org\/10.2307\/2273546","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,6]]}}}