{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T18:59:58Z","timestamp":1699037998831},"reference-count":10,"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":15351,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,3]]},"abstract":"<jats:p>In [10, \u00a70, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory <jats:italic>T<\/jats:italic> is \u2135<jats:sub>0<\/jats:sub>-stable (totally transcendental) and not \u2135<jats:sub>1<\/jats:sub>-categorical, then it has at least \u22231 + <jats:italic>\u03b1<\/jats:italic>\u2223 models of power \u2135<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>.<\/jats:p><jats:p>In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of \u2135<jats:sub>0<\/jats:sub>-stability and \u2135<jats:sub>1<\/jats:sub>-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].<\/jats:p><jats:p>Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200080646_inline1\" \/> of power \u2135<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> such that there is a countable definable subset in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200080646_inline2\" \/>. Let <jats:italic>B<\/jats:italic><jats:sub>0<\/jats:sub> be such a subset. Say <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200080646_inline3\" \/>. We will give by transfinite induction an elementary chain <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200080646_inline4\" \/> of models of <jats:italic>T<\/jats:italic> of power \u2135<jats:sub>\u03b1<\/jats:sub> such that B[i<jats:sub>1<\/jats:sub> \u2026 i<jats:italic><jats:sub>n<\/jats:sub><\/jats:italic>]<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200080646_inline5\" \/> has power \u2135<jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub> and such that every infinite definable subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200080646_inline6\" \/> has power \u2265\u2135<jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub>. This clearly suffices.<\/jats:p>","DOI":"10.2307\/2272555","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:14:18Z","timestamp":1146935658000},"page":"133-134","source":"Crossref","is-referenced-by-count":5,"title":["A new proof of a theorem of Shelah"],"prefix":"10.1017","volume":"37","author":[{"given":"John W.","family":"Rosenthal","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200080646_ref010","unstructured":"Shelah S. , Stability, the F.c.p. and super stability; model theoretic properties of formulas in first order theory (to appear)."},{"key":"S0022481200080646_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90007-0"},{"key":"S0022481200080646_ref008","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200080646_ref006","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10648"},{"key":"S0022481200080646_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/1970549"},{"key":"S0022481200080646_ref002","volume-title":"Models and ultraproducts","author":"Bell","year":"1969"},{"key":"S0022481200080646_ref001","first-page":"79","volume":"36","author":"Baldwin","year":"1971","journal-title":"On strongly minimal sets"},{"key":"S0022481200080646_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"S0022481200080646_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(64)50021-9"},{"key":"S0022481200080646_ref007","volume-title":"Constructible sets with applications","author":"Mostowski","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\/S0022481200080646","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T15:19:12Z","timestamp":1559315952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200080646\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1972,3]]}},"alternative-id":["S0022481200080646"],"URL":"https:\/\/doi.org\/10.2307\/2272555","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,3]]}}}