{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T06:47:34Z","timestamp":1648968454768},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>In this paper we investigate omitting types for a certain kind of stable theories which we call stable ccc theories. In Theorem 2.1 we improve Steinhorn's result from [St]. We prove also some independence results concerning omitting types. The main results presented in this paper were part of the author's Ph.D. thesis [N1].<\/jats:p><jats:p>Throughout, we use the standard set-theoretic and model-theoretic notation, such as can be found for example in [Sh] or [M]. So in particular <jats:italic>T<\/jats:italic> is always a <jats:italic>countable<\/jats:italic> complete theory in the language <jats:italic>L<\/jats:italic>. We consider all models of <jats:italic>T<\/jats:italic> and all sets of parameters subsets of the monster model \u212d, which is very saturated. <jats:italic>L<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>A<\/jats:italic>) denotes the Lindenbaum-Tarski algebra of formulas with parameters from <jats:italic>A<\/jats:italic> and <jats:italic>n<\/jats:italic> free variables. We omit <jats:italic>n<\/jats:italic> in <jats:italic>L<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>A<\/jats:italic>) when <jats:italic>n<\/jats:italic> = 1 or when it is clear from the context what <jats:italic>n<\/jats:italic> is. If <jats:italic>\u03c6, \u03c8<\/jats:italic> \u2208 <jats:italic>L<\/jats:italic>(<jats:italic>A<\/jats:italic>) are consistent then we say that <jats:italic>\u03c6<\/jats:italic> is <jats:italic>below \u03c8<\/jats:italic> if <jats:italic>\u03c8<\/jats:italic>\u22a2<jats:italic>\u03c8<\/jats:italic>. For a type <jats:italic>p<\/jats:italic> and a set <jats:italic>A<\/jats:italic> \u2286 \u212d, <jats:italic>p<\/jats:italic>(<jats:italic>A)<\/jats:italic> is the set of tuples of elements of <jats:italic>A<\/jats:italic> which satisfy <jats:italic>p<\/jats:italic>. Formulas are special cases of types. We say that a type <jats:italic>p<\/jats:italic> is <jats:italic>isolated<\/jats:italic> over <jats:italic>A<\/jats:italic> if, for some <jats:italic>\u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025548_inline3\" \/>)<\/jats:italic> \u2208 <jats:italic>L(A)<\/jats:italic>, <jats:italic>\u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025548_inline3\" \/>)<\/jats:italic> \u22a2 <jats:italic>p(x)<\/jats:italic>, i.e. <jats:italic>\u03c6<\/jats:italic> isolates <jats:italic>p<\/jats:italic>. For a formula <jats:italic>\u03c6<\/jats:italic>, [<jats:italic>\u03c6<\/jats:italic>] denotes the class of types which contain <jats:italic>\u03c6<\/jats:italic>. We assume that the reader is familiar with some basic knowledge of forking, as presented in [Sh, III] or [M].<\/jats:p><jats:p>Throughout, we work in ZFC. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025548_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025548_inline2\" \/> denote (countable) transitive models of ZFC. cov <jats:bold>K<\/jats:bold> is the minimal number of meager sets covering the real line <jats:bold>R<\/jats:bold>. In this paper we prove theorems showing connections between omitting types and the combinatorics of the real line. More results in this direction are presented in [N2] and [N3].<\/jats:p>","DOI":"10.2307\/2274472","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:36:50Z","timestamp":1146955010000},"page":"1037-1047","source":"Crossref","is-referenced-by-count":1,"title":["Omitting types for stable ccc theories"],"prefix":"10.1017","volume":"55","author":[{"given":"Ludomir","family":"Newelski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025548_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1983-0715871-9"},{"key":"S0022481200025548_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-77-1-9-20"},{"key":"S0022481200025548_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760649"},{"key":"S0022481200025548_ref008","volume-title":"Classification theory and the number of non-isomorphic types","author":"Shelah","year":"1978"},{"key":"S0022481200025548_ref006","first-page":"1020","volume":"52","author":"Newelski","year":"1987","journal-title":"Omitting types and the real line"},{"key":"S0022481200025548_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067645"},{"key":"S0022481200025548_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19750210108"},{"key":"S0022481200025548_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02788174"},{"key":"S0022481200025548_ref005","unstructured":"Newelski L. , Ph.D. thesis, Institute of Mathematics, Polish Academy of Sciences, Wroc\u0142aw, 1986."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025548","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:25:01Z","timestamp":1558211101000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025548\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025548"],"URL":"https:\/\/doi.org\/10.2307\/2274472","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}