{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,8]],"date-time":"2024-02-08T11:10:14Z","timestamp":1707390614817},"reference-count":11,"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":5124,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2000,3]]},"abstract":"Abstract<\/jats:title>Suppose \u03bb is a singular cardinal of uncountable cofinality \u03ba. For a modelof cardinality \u03bb, let No() denote the number of isomorphism types of modelsof cardinality \u03bb which areL<\/jats:italic>\u221e\u03bb<\/jats:sub>-equivalent to. In [7] Shelah considered inverse \u03ba-systemsof abelian groups and their certain kind of quotient limits Gr()\/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal \u039c there exists an inverse \u03ba-systemsuch thatconsists of abelian groups having cardinality at most \u039c\u03ba<\/jats:sup>and card(Gr()\/ Fact()) = \u039c. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse \u03ba-systems and possible values of No (under the assumption that \u03b8\u03ba<\/jats:sup>< \u03bb for every \u03b8 < \u03bb): ifis an inverse \u03ba-system of abelian groups having cardinality < \u03bb, then there is a modelsuch that card() = \u03bb and No() = card(Gr()\/ Fact()). The following was an immediate consequence (when \u03b8\u03ba<\/jats:sup>< \u03bb for every \u03b8 < \u03bb): for every nonzero \u039c < \u03bb or \u039c = \u03bb\u03ba<\/jats:sup>there is a model, of cardinality \u03bb with No() = \u039c. In this paper we show: for every nonzero \u039c \u2264 \u03bb\u03ba<\/jats:sub>there is an inverse \u03ba-systemof abelian groups having cardinality < \u03bb such that card(Gr()\/ Fact()) = \u039c (under the assumptions 2\u03ba<\/jats:sup>< \u03bb and \u03b8<\u03ba<\/jats:sup>< \u03bb for all \u03b8 < \u03bb when \u039c > \u03bb), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = \u03bb is possible when \u03b8\u03ba<\/jats:sup>> \u03bb for every \u03bb < \u03bb.<\/jats:p>","DOI":"10.2307\/2586536","type":"journal-article","created":{"date-parts":[[2006,11,18]],"date-time":"2006-11-18T09:44:27Z","timestamp":1163843067000},"page":"272-284","source":"Crossref","is-referenced-by-count":0,"title":["On inverse \u03b3-systems and the number ofL<\/i>\u221e\u03bb<\/sub>-equivalent, non-isomorphic models for \u03bb singular"],"prefix":"10.1017","volume":"65","author":[{"given":"Saharon","family":"Shelah","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pauli","family":"V\u00e4is\u00e4nen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200012469_ref009","first-page":"1431","volume":"54","author":"Shelah","year":"1989","journal-title":"The number ofpairwisenon-elementarily-embeddable models"},{"key":"S0022481200012469_ref006","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093883562"},{"key":"S0022481200012469_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01669282"},{"key":"S0022481200012469_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079681"},{"key":"S0022481200012469_ref002","first-page":"443","article-title":"Number of models in L\u221e,\u03c9 theories, II","volume":"16","author":"Palyutin","year":"1977","journal-title":"Algebra i Logika"},{"key":"S0022481200012469_ref004","first-page":"329","volume-title":"Theory of models (Proceedings of the 1963 International Symposium, Berkeley)","author":"Scott","year":"1965"},{"key":"S0022481200012469_ref005","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093883334"},{"key":"S0022481200012469_ref007","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093870759"},{"key":"S0022481200012469_ref010","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537854.001.0001","volume-title":"Cardinal arithmetic","author":"Shelah","year":"1994"},{"key":"S0022481200012469_ref011","volume-title":"Transactions of the American Mathematical Society","author":"Shelah"},{"key":"S0022481200012469_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0098507"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200012469","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,8]],"date-time":"2024-02-08T10:34:48Z","timestamp":1707388488000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200012469\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,3]]}},"alternative-id":["S0022481200012469"],"URL":"https:\/\/doi.org\/10.2307\/2586536","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,3]]}}}