{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T17:38:29Z","timestamp":1775497109778,"version":"3.50.1"},"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":13068,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,6]]},"abstract":"<jats:p>It is the purpose of this paper to investigate the model theory of logic with a generalized quantifier; in particular the logic <jats:italic>L(Q<jats:sub>1<\/jats:sub>)<\/jats:italic> where <jats:italic>Q<jats:sub>1<\/jats:sub>x\u03c6(x)<\/jats:italic> has the intended meaning \u201cthere exist uncountably many <jats:italic>x<\/jats:italic> such that \u03c6(<jats:italic>x<\/jats:italic>)\u201d. We do this from the point of view that the best way to study what happens in the so-called \u201c\u03c9<jats:sub>1<\/jats:sub>-standard\u201d models of <jats:italic>L(Q<jats:sub>1<\/jats:sub>)<\/jats:italic> is to examine the countable ideal models of <jats:italic>L(Q)<\/jats:italic> that satisfy all of the axioms for <jats:italic>L(Q<jats:sub>1<\/jats:sub>)<\/jats:italic> (see definitions of \u03c9<jats:sub>1<\/jats:sub>-standard and ideal models in \u00a71). We believe that this study can be as fruitful for <jats:italic>L(Q<jats:sub>1<\/jats:sub>)<\/jats:italic> as the study of countable models of ZF has been for set theory.<\/jats:p><jats:p>A major problem is formulating an adequate definition of submodel for countable ideal models that is compatible with that for \u03c9<jats:sub>1<\/jats:sub>-standard models. Thus we begin the paper by discussing several possible definitions of the notion of submodel. We then adopt a particular definition of submodel and investigate model-completeness in <jats:italic>L(Q)<\/jats:italic>. We define model-completeness both for \u03c9<jats:sub>1<\/jats:sub>-standard models and for countable ideal models and compare the two notions. We also examine elimination of quantifiers, as well as investigating formulas preserved under submodels, again both for \u03c9<jats:sub>1<\/jats:sub>-standar d and countable ideal models.<\/jats:p>","DOI":"10.2307\/2272829","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:47:35Z","timestamp":1146952055000},"page":"304-321","source":"Crossref","is-referenced-by-count":10,"title":["Ideal models and some not so ideal problems in the model theory of <i>L(Q)<\/i>"],"prefix":"10.1017","volume":"43","author":[{"given":"Kim B.","family":"Bruce","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049653_ref019","unstructured":"Shelah S. [1976], Personal communication."},{"key":"S0022481200049653_ref015","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1972-12921-5"},{"key":"S0022481200049653_ref013","first-page":"117","article-title":"On completeness in cardinality logics","volume":"23","author":"Jensen","year":"1975","journal-title":"Bulletin de L'Academie Polonaise des Sciences"},{"key":"S0022481200049653_ref009","first-page":"205","volume-title":"Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics XXV","author":"Feferman","year":"1974"},{"key":"S0022481200049653_ref007","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200049653_ref006","unstructured":"Bruce K. B. [1978a], Model constructions in stationary logic, Part I: Forcing (to appear)."},{"key":"S0022481200049653_ref005","volume-title":"Annals of Mathematical Logic","author":"Bruce","year":"1978"},{"key":"S0022481200049653_ref001","volume-title":"Annals of Mathematical Logic","author":"Barwise","year":"1977"},{"key":"S0022481200049653_ref002","first-page":"531","volume":"41","author":"Barwise","year":"1976","journal-title":"An introduction to recursively saturated and resplendent models"},{"key":"S0022481200049653_ref021","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1975.56.265"},{"key":"S0022481200049653_ref017","volume-title":"Introduction to model theory and to the metamathematics of algebra","author":"Robinson","year":"1974"},{"key":"S0022481200049653_ref016","volume-title":"Annals of Mathematical cogic","author":"Magidor","year":"1977"},{"key":"S0022481200049653_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(70)80005-5"},{"key":"S0022481200049653_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90014-4"},{"key":"S0022481200049653_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762675"},{"key":"S0022481200049653_ref008","unstructured":"COWLES J. [1975], Abstract logic and extensions of first-order logic, Ph.D. Thesis, Pennsylvania State University."},{"key":"S0022481200049653_ref004","unstructured":"Bruce K. B. [1975], Model-theoretic forcing with a generalized quantifier, Ph.D. Thesis, University of Wisconsin."},{"key":"S0022481200049653_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF02834760"},{"key":"S0022481200049653_ref020","unstructured":"Shelah S. [1977], Personal communication."},{"key":"S0022481200049653_ref018","first-page":"183","volume":"43","author":"Schlipf","year":"1978","journal-title":"Toward model theory through recursive saturation"},{"key":"S0022481200049653_ref003","volume-title":"Models and ultraproducts","author":"Bell","year":"1969"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049653","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:57:48Z","timestamp":1558987068000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049653\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049653"],"URL":"https:\/\/doi.org\/10.2307\/2272829","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}