{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T12:09:27Z","timestamp":1768738167153,"version":"3.49.0"},"reference-count":23,"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":557,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>\u03ba<\/jats:italic> be an infinite cardinal. A subset of (<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic>)<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> is a <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline01\"\/>-subset if it is the projection <jats:italic>p<\/jats:italic>[T] of all cofinal branches through a subtree <jats:italic>T<\/jats:italic> of (<jats:sup>&gt;<jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic>)<jats:sup><jats:italic>n+1<\/jats:italic><\/jats:sup> of height <jats:italic>\u03ba<\/jats:italic>. We define <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline02\"\/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline03\"\/>-subsets of (<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic>)<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> as usual.<\/jats:p><jats:p>Given an uncountable regular cardinal <jats:italic>\u03ba<\/jats:italic> with <jats:italic>\u03ba<\/jats:italic> = <jats:italic>\u03ba<\/jats:italic><jats:sup>&lt;<jats:italic>\u03ba<\/jats:italic><\/jats:sup> and an arbitrary subset <jats:italic>A<\/jats:italic> of <jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic>, we show that there is a &lt;<jats:italic>\u03ba<\/jats:italic>-closed forcing \u2119 that satisfies the <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-chain condition and forces <jats:italic>A<\/jats:italic> to be a <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline01\"\/>-subset of <jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic> in every \u2119-generic extension of V. We give some applications of this result and the methods used in its proof.<\/jats:p><jats:p>(i) Given any set <jats:italic>x<\/jats:italic>, we produce a partial order with the above properties that forces <jats:italic>x<\/jats:italic> to be an element of <jats:italic>L<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline04\"\/>.<\/jats:p><jats:p>(ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of <jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic> whose graph is a <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline05\"\/>-subset of <jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic> \u00d7 <jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic>.<\/jats:p><jats:p>(iii) We provide a short proof of a result due to Mekler and V\u00e4\u00e4n\u00e4nen by using the above forcing to add a tree <jats:italic>T<\/jats:italic> of cardinality and height <jats:italic>\u03ba<\/jats:italic> such that <jats:italic>T<\/jats:italic> has no cofinal branches and every tree from the ground model of cardinality and height <jats:italic>\u03ba<\/jats:italic> without a cofinal branch quasi-order embeds into <jats:italic>T<\/jats:italic>.<\/jats:p><jats:p>(iv) We will show that generic absoluteness for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline06\"\/>-formulae (i.e., formulae with parameters which define <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline07\"\/>-subsets of <jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup><jats:italic>\u03ba<\/jats:italic>) under &lt;<jats:italic>\u03ba<\/jats:italic>-closed forcings that satisfy the <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-chain condition is inconsistent.<\/jats:p><jats:p>In another direction, we use methods from the proofs of the above results to show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline01\"\/>- and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000608_inline08\"\/>-subsets have some useful structural properties in certain ZFC-models.<\/jats:p>","DOI":"10.2178\/jsl\/1344862172","type":"journal-article","created":{"date-parts":[[2012,11,5]],"date-time":"2012-11-05T10:02:32Z","timestamp":1352109752000},"page":"1011-1046","source":"Crossref","is-referenced-by-count":14,"title":["-definability at uncountable regular cardinals"],"prefix":"10.1017","volume":"77","author":[{"given":"Philipp","family":"L\u00fccke","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200000608_ref015","volume-title":"The higher infinite","author":"Kanamori","year":"2003"},{"key":"S0022481200000608_ref020","doi-asserted-by":"publisher","DOI":"10.4064\/fm174-2-1"},{"key":"S0022481200000608_ref013","first-page":"1","volume":"36","author":"Jech","year":"1971","journal-title":"Trees"},{"key":"S0022481200000608_ref019","first-page":"281","article-title":"Proper forcings and absoluteness in 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