{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T08:56:48Z","timestamp":1768985808015,"version":"3.49.0"},"reference-count":13,"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":3479,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,9]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>We study the preservation under projective ccc forcing extensions of the property of <jats:italic>L<\/jats:italic>(\u211d) being a Solovay model. We prove that this property is preserved by every strongly-<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000760X_inline1\"\/> absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000760X_inline2\"\/> absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.<\/jats:p>","DOI":"10.2178\/jsl\/1096901764","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T16:42:13Z","timestamp":1109781733000},"page":"742-766","source":"Crossref","is-referenced-by-count":8,"title":["Solovay models and forcing extensions"],"prefix":"10.1017","volume":"69","author":[{"given":"Joan","family":"Bagaria","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roger","family":"Bosch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000760X_ref010","volume-title":"Perspectives in Mathematical Logic","author":"Kanamori","year":"1994"},{"key":"S002248120000760X_ref008","first-page":"374","volume":"60","author":"Judah","year":"1995","journal-title":"Martin Axiom and the size of the continuum"},{"key":"S002248120000760X_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(00)00038-5"},{"key":"S002248120000760X_ref005","volume-title":"Set theory of the continuum","author":"Bagaria","year":"1992"},{"key":"S002248120000760X_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF02801471"},{"key":"S002248120000760X_ref009","first-page":"1188","volume":"53","author":"Judah","year":"1988","journal-title":"Souslin forcing"},{"key":"S002248120000760X_ref001","unstructured":"Bagaria J. , Definable forcing and regularity properties of projective sets of reals, Ph.D. thesis , University of California at Berkeley, 1991."},{"key":"S002248120000760X_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(96)00035-8"},{"key":"S002248120000760X_ref003","unstructured":"Bagaria J. and Bosch R. , Proper forcing extensions and Solovay models, to appear."},{"key":"S002248120000760X_ref006","volume-title":"Set Theory","author":"Jech","year":"2003"},{"key":"S002248120000760X_ref007","volume-title":"Mathematical Logic and Foundations of Set Theory","author":"Jensen","year":"1970"},{"key":"S002248120000760X_ref012","doi-asserted-by":"publisher","DOI":"10.2307\/1970696"},{"key":"S002248120000760X_ref013","volume-title":"Logic Colloquium'81","author":"Woodin","year":"1982"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000760X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T16:27:22Z","timestamp":1557160042000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000760X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2004,9]]}},"alternative-id":["S002248120000760X"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1096901764","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,9]]}}}