{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T06:10:46Z","timestamp":1683094246268},"reference-count":14,"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":9507,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present a downward L\u00f6wenheim-Skolem theorem which transfers downward formulas from <jats:italic>L<jats:sub>\u221e,\u03c9<\/jats:sub><\/jats:italic> to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200029066_inline1\" \/>, <jats:italic>\u03c9<\/jats:italic>. The simplest instance is:<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 1. <jats:italic>Let <jats:italic>\u03bb<\/jats:italic> &gt; \u03ba be infinite cardinals, and let L be a similarity type of cardinality \u03ba at most. For every L-structure M of cardinality <jats:italic>\u03bb<\/jats:italic> and every X<\/jats:italic> \u2286 <jats:italic>M there exists a model N \u227a M containing the set X of power<\/jats:italic> \u2223<jats:italic>X<\/jats:italic>\u2223 \u00b7 \u03ba <jats:italic>such that for every pair of finite sequences<\/jats:italic> a, b \u2208 <jats:italic>N<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200029066_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>The following theorem is an application:<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 2. <jats:italic>Let \u03bb&lt;\u03ba, T<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200029066_inline1\" \/>, <jats:italic>\u03c9, and suppose \u03c7 is a Ramsey cardinal greater than \u03bb. If T has the<\/jats:italic> (<jats:italic>\u03c7<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200029066_inline1\" \/>, <jats:italic>\u03c9)-unsuperstability property, then T has the<\/jats:italic> (<jats:italic>\u03c7<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200029066_inline1\" \/>, <jats:italic>\u03c9)-unsuperstability property<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s0022481200029066","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T12:41:05Z","timestamp":1394714465000},"page":"231-242","source":"Crossref","is-referenced-by-count":0,"title":["A downward L\u00f6wenheim-Skolem theorem for infinitary theories which have the unsuperstability property"],"prefix":"10.1017","volume":"53","author":[{"given":"Rami","family":"Grossberg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200029066_ref005","first-page":"167","volume-title":"Infinitistic methods: proceedings of the symposium on foundations of mathematics","author":"Henkin","year":"1961"},{"key":"S0022481200029066_ref006","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200029066_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079681"},{"key":"S0022481200029066_ref004","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1215\/ijm\/1256044645","article-title":"A non structure theorem for an infinitary theory which has the unsuperstability property","volume":"30","author":"Grossberg","year":"1986","journal-title":"Illinois Journal of Mathematics"},{"key":"S0022481200029066_ref011","unstructured":"[Ma] Malitz J. , Problems in the model theory of infinite languages, Ph.D. thesis, University of California, Berkeley, California, 1966."},{"key":"S0022481200029066_ref007","first-page":"405","volume-title":"The theory of models","author":"Karp","year":"1965"},{"key":"S0022481200029066_ref014","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200029066_ref003","first-page":"245","volume-title":"Contributions to the theory of games. 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