{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T13:59:57Z","timestamp":1649167197763},"reference-count":20,"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":9142,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>L<\/jats:italic>(<jats:bold>Q<\/jats:bold>) be first order logic with Keisler's quantifier, in the \u03bb<jats:sup>+<\/jats:sup> interpretation (= the satisfaction is defined as follows: <jats:italic>M<\/jats:italic> \u22a8 (<jats:bold>Q<\/jats:bold><jats:italic>x<\/jats:italic>)<jats:sub><jats:italic>\u03c6<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) means there are <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup> many elements in <jats:italic>M<\/jats:italic> satisfying the formula <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>)).<\/jats:p><jats:p>Theorem 1. <jats:italic>Let <jats:italic>\u03bb<\/jats:italic> be a singular cardinal; assume <jats:italic>\u25a1<jats:sub>\u03bb<\/jats:sub><\/jats:italic> and GCH. If T is a complete theory in L<\/jats:italic>(<jats:bold>Q<\/jats:bold>) <jats:italic>of cardinality at most \u03bb, and p is an L<\/jats:italic>(<jats:bold>Q<\/jats:bold>) 1-<jats:italic>type so that T strongly omits p( = p has no support, to be defined in \u00a71), then T has a model of cardinality <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup> in the <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup> interpretation which omits p<\/jats:italic>.<\/jats:p><jats:p>Theorem 2. <jats:italic>Let \u03bb be a singular cardinal, and let T be a complete first order theory of cardinality \u03bb at most. Assume \u25a1<jats:sub>\u03bb<\/jats:sub> and GCH. If \u0393 is a smallness notion then T has a model of cardinality \u03bb<jats:sup>+<\/jats:sup> such that a formula \u03c6(<jats:italic>x<\/jats:italic>) is realized by \u03bb<jats:sup>+<\/jats:sup> elements of M iff \u03c6(<jats:italic>x<\/jats:italic>) is not \u0393-small. The theorem is proved also when \u03bb is regular assuming \u03bb = \u03bb<jats:sup>&lt;\u03bb<\/jats:sup>. It is new when \u03bb is singular or when \u2223T\u2223 = \u03bb is regular<\/jats:italic>.<\/jats:p><jats:p>Theorem 3. <jats:italic>Let \u03bb be singular. If<\/jats:italic> Con(<jats:italic>ZFC + GCH + \u2203\u03ba) [\u03ba is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275020","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:29:26Z","timestamp":1146940166000},"page":"122-137","source":"Crossref","is-referenced-by-count":0,"title":["Models with second order properties in successors of singulars"],"prefix":"10.1017","volume":"54","author":[{"given":"Rami","family":"Grossberg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200027638_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90008-6"},{"key":"S0022481200027638_ref003","first-page":"1029","volume":"51","author":"Ben-David","year":"1986","journal-title":"The weak \u25a1* is really weaker than the full \u25a1"},{"key":"S0022481200027638_ref002","first-page":"229","article-title":"On the strength of gap one principle for successors of singulars","volume":"5","author":"Ben-David","year":"1984","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200027638_ref009","first-page":"592","article-title":"Omitting types theorem for L(Q)","volume":"3","author":"Grossberg","year":"1982","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200027638_ref019","article-title":"Gap one two-cardinal principle and the omitting type theorem for L(Q)","author":"Shelah","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200027638_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0059290"},{"key":"S0022481200027638_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90009-8"},{"key":"S0022481200027638_ref013","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200027638_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761504"},{"key":"S0022481200027638_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1965-0193016-3"},{"key":"S0022481200027638_ref018","volume-title":"Annals of Pure and Applied Logic","author":"Shelah"},{"key":"S0022481200027638_ref005","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200027638_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF02011630"},{"key":"S0022481200027638_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(70)80005-5"},{"key":"S0022481200027638_ref008","first-page":"663","volume":"41","author":"Gregory","year":"1976","journal-title":"Higher Souslin trees and the generalized continuum hypothesis"},{"key":"S0022481200027638_ref010","first-page":"199","article-title":"Chang Shelah two cardinal theorem for successors of singulars","volume":"5","author":"Grossberg","year":"1984","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200027638_ref012","unstructured":"Litman A. , Ph.D. thesis, The Hebrew University, Jerusalem, 1982. (Hebrew; English abstract)"},{"key":"S0022481200027638_ref007","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600586"},{"key":"S0022481200027638_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90013-1"},{"key":"S0022481200027638_ref020","first-page":"390","volume-title":"Theory of models","author":"Vaught","year":"1965"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200027638","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T17:01:48Z","timestamp":1558285308000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027638\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,3]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1989,3]]}},"alternative-id":["S0022481200027638"],"URL":"https:\/\/doi.org\/10.2307\/2275020","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,3]]}}}