{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T08:36:03Z","timestamp":1649061363067},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12885,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,12]]},"abstract":"<jats:p>We shall describe Boolean extensions of models of set theory with the axiom of choice in which cardinals are collapsed by mappings definable from parameters in the ground model. In particular, starting from the constructible universe, we get Boolean extensions in which constructible cardinals are collapsed by ordinal definable sets.<\/jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/> be a transitive model of set theory with the axiom of choice. Definability of sets in the generic extensions of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/> is closely related to the automorphisms of the corresponding Boolean algebra. In particular, if <jats:italic>G<\/jats:italic> is an <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>-generic ultrafilter on a rigid complete Boolean algebra <jats:italic>C<\/jats:italic>, then every set in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>[<jats:italic>G<\/jats:italic>] is definable from parameters in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>. Hence if <jats:italic>B<\/jats:italic> is a complete Boolean algebra containing a set of forcing conditions to collapse some cardinals in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>, it suffices to construct a rigid complete Boolean algebra <jats:italic>C<\/jats:italic>, in which <jats:italic>B<\/jats:italic> is completely embedded. If <jats:italic>G<\/jats:italic> is as above, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>[<jats:italic>G<\/jats:italic>] satisfies \u201cevery set is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>-definable\u201d and the inner model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>[<jats:italic>G<\/jats:italic> \u2229 <jats:italic>B<\/jats:italic>] contains the collapsing mapping determined by <jats:italic>B<\/jats:italic>. To complete the result, it is necessary to give some conditions under which every cardinal from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>[<jats:italic>G<\/jats:italic> \u2229 <jats:italic>B<\/jats:italic>] remains a cardinal in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049148_inline1\" \/>[<jats:italic>G<\/jats:italic>].<\/jats:p><jats:p>The absolutness is granted for every cardinal at least as large as the saturation of <jats:italic>C<\/jats:italic>. To keep the upper cardinals absolute, it often suffices to construct <jats:italic>C<\/jats:italic> with the same saturation as <jats:italic>B<\/jats:italic>. It was shown in [6] that this is always possible, namely, that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra with the same saturation.<\/jats:p>","DOI":"10.2307\/2273502","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:57Z","timestamp":1146937737000},"page":"635-642","source":"Crossref","is-referenced-by-count":2,"title":["Cardinal collapsing and ordinal definability"],"prefix":"10.1017","volume":"43","author":[{"given":"Petr","family":"\u0160t\u011bp\u00e1nek","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049148_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90005-2"},{"key":"S0022481200049148_ref002","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-35.1.85"},{"key":"S0022481200049148_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758125"},{"key":"S0022481200049148_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-57-3-275-285"},{"key":"S0022481200049148_ref006","first-page":"64","volume":"42","author":"\u0160t\u011bp\u00e1nek","year":"1977","journal-title":"Embedding theorems for Boolean algebras and consistency results on ordinal definable sets"},{"key":"S0022481200049148_ref007","volume-title":"The theory of semisets","author":"Vop\u011bnka","year":"1972"},{"key":"S0022481200049148_ref004","first-page":"553","volume-title":"Proceedings of the International Congress of Mathematicians, Vancouver, 1974","author":"Shelah","year":"1975"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049148","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:07:52Z","timestamp":1558969672000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049148\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1978,12]]}},"alternative-id":["S0022481200049148"],"URL":"https:\/\/doi.org\/10.2307\/2273502","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,12]]}}}