{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T12:53:26Z","timestamp":1681217606515},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log"],"published-print":{"date-parts":[[2012,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Vop\u011bnka proved long ago that every set of ordinals is set-generic over <jats:bold>HOD<\/jats:bold>, G\u00f6del's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe <jats:italic>V<\/jats:italic> is class-generic over (<jats:bold>HOD,S<\/jats:bold>), and indeed over the even smaller inner model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600000147_inline1\" \/> =(<jats:italic>L<\/jats:italic>[<jats:italic>S<\/jats:italic>],<jats:italic>S<\/jats:italic>), where <jats:italic>S<\/jats:italic> is the <jats:italic>Stability predicate<\/jats:italic>. We refer to the inner model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600000147_inline1\" \/> as the <jats:italic>Stable Core of V<\/jats:italic>. The predicate <jats:italic>S<\/jats:italic> has a simple definition which is more absolute than any definition of <jats:bold>HOD<\/jats:bold>; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core (this is not possible for <jats:bold>HOD<\/jats:bold> by Vop\u011bnka's theorem).<\/jats:p>","DOI":"10.2178\/bsl\/1333560807","type":"journal-article","created":{"date-parts":[[2012,4,4]],"date-time":"2012-04-04T14:17:21Z","timestamp":1333549041000},"page":"261-267","source":"Crossref","is-referenced-by-count":3,"title":["The Stable core"],"prefix":"10.1017","volume":"18","author":[{"given":"Sy-david","family":"Friedman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600000147_ref001","doi-asserted-by":"crossref","unstructured":"Friedman S. , Fine structure and class forcing , de Gruyter Series in Logic and its Applications, vol. 3, de Gruyter, 2000.","DOI":"10.1515\/9783110809114"},{"key":"S1079898600000147_ref002","volume-title":"The theory of semisets","author":"Vop\u011bnka","year":"1972"}],"container-title":["The Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600000147","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T16:45:29Z","timestamp":1556210729000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600000147\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6]]},"references-count":2,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,6]]}},"alternative-id":["S1079898600000147"],"URL":"https:\/\/doi.org\/10.2178\/bsl\/1333560807","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,6]]}}}