{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T21:28:50Z","timestamp":1695850130624},"reference-count":10,"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:title>Abstract<\/jats:title><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline2\" \/> be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal <jats:italic>\u03b1<\/jats:italic> with the property that<\/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=\"S0022481200025457_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>i.e. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline2\" \/> are <jats:italic>L<jats:sub>\u221e\u03c9<\/jats:sub><\/jats:italic>-equivalent up to quantifier-rank <jats:italic>\u03b1<\/jats:italic> but not up to <jats:italic>\u03b1<\/jats:italic> + 1. In this paper we consider models <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline2\" \/> of cardinality <jats:italic>\u03c9<jats:sub>1<\/jats:sub><\/jats:italic> and construct trees which have a similar relation to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline2\" \/> as a above. For this purpose we introduce a new ordering <jats:italic>T<\/jats:italic> \u226a <jats:italic>T\u2032<\/jats:italic> of trees, which may have some independent interest of its own. It turns out that the above ordinal <jats:italic>\u03b1<\/jats:italic> has two qualities which coincide in countable models but will differ in uncountable models. Respectively, two kinds of trees emerge from <jats:italic>\u03b1<\/jats:italic>. We call them Scott trees and Karp trees, respectively. The definition and existence of these trees is based on an examination of the Ehrenfeucht game of length <jats:italic>\u03c9<jats:sub>1<\/jats:sub><\/jats:italic> between <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline2\" \/>. We construct two models of power <jats:italic>\u03c9<jats:sub>1<\/jats:sub><\/jats:italic> with <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025457_inline3\" \/> mutually noncomparable Scott trees.<\/jats:p>","DOI":"10.2307\/2274463","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:36:50Z","timestamp":1146955010000},"page":"897-908","source":"Crossref","is-referenced-by-count":26,"title":["On Scott and Karp trees of uncountable models"],"prefix":"10.1017","volume":"55","author":[{"given":"Tapani","family":"Hyttinen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jouko","family":"V\u00e4\u00e4n\u00e4nen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025457_ref007","article-title":"Syntax and semantics of infinitely deep languages","volume":"50","author":"Karttunen","year":"1984","journal-title":"Annales Academiae Scientiarum Fennicae Series A I: Mathematica Dissertationes"},{"key":"S0022481200025457_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200025457_ref004","article-title":"Games and infinitary languages","volume":"64","author":"Hyttinen","year":"1987","journal-title":"Annales Academiae Scientiarum Fennicae Series A I: Mathematica Dissertationes"},{"key":"S0022481200025457_ref002","first-page":"481\u2013482","volume":"41","author":"Baumgartner","year":"1976","journal-title":"Adding a closed unbounded set"},{"key":"S0022481200025457_ref006","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200025457_ref010","first-page":"249\u2013262","article-title":"Stationary sets, trees and continuums","volume":"29","author":"Todor\u010devi\u0107","year":"1981","journal-title":"Publications de l' institut Math\u00e9matique"},{"key":"S0022481200025457_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90026-7"},{"key":"S0022481200025457_ref009","first-page":"51\u201374","article-title":"L\u221e\u03bb-equivalence, isomorphism and potential isomorphism","volume":"236","author":"Nadel","year":"1978","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200025457_ref005","volume-title":"Fundamenta Mathematicae","author":"Hyttinen"},{"key":"S0022481200025457_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90010-9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025457","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:24:41Z","timestamp":1558211081000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025457\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025457"],"URL":"https:\/\/doi.org\/10.2307\/2274463","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}