{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T22:52:10Z","timestamp":1654987930844},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3663,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,3]]},"abstract":"In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.<\/jats:p>The statement \u201c absoluteness holds for ccc forcing\u201d means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V<\/jats:italic>, then it already has a solution in V<\/jats:italic>. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.<\/jats:p>Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC<\/jats:italic> + absoluteness for ccc<\/jats:italic>.<\/jats:p>The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].<\/jats:p>Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a<\/jats:italic> reflecting cardinal, i.e., a regular cardinal \u03ba such that H<\/jats:italic>(\u03ba<\/jats:italic>) is<\/jats:italic> \u22112<\/jats:sub>-elementary in V<\/jats:italic>.<\/jats:p>(b) absoluteness for class-forcing is inconsistent<\/jats:italic>.<\/jats:p>We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and \u03c91<\/jats:sub>-preserving. We refer the reader to [8] for the definitions of these forcing notions.<\/jats:p>Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L<\/jats:italic> either \u03c91<\/jats:sub> is Mahlo or \u03c92<\/jats:sub> is inaccessible).<\/jats:p>","DOI":"10.2178\/jsl\/1080938826","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:25:33Z","timestamp":1109798733000},"page":"73-80","source":"Crossref","is-referenced-by-count":2,"title":["Generic \u03a33<\/sub>1<\/sup> absoluteness"],"prefix":"10.1017","volume":"69","author":[{"given":"Sy D.","family":"Friedman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000801X_ref007","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093870823"},{"key":"S002248120000801X_ref008","volume-title":"Set Theory","author":"Jech","year":"2002"},{"key":"S002248120000801X_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-2776-5_12"},{"key":"S002248120000801X_ref006","first-page":"58","volume":"60","author":"Goldstern","year":"1995","journal-title":"The bounded proper forcing axiom"},{"key":"S002248120000801X_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-00-02636-2"},{"key":"S002248120000801X_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9754-0_21"},{"key":"S002248120000801X_ref001","first-page":"366","volume":"62","author":"Bagaria","year":"1997","journal-title":"A characterization of Martin's axiom in terms of absoluteness"},{"key":"S002248120000801X_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(00)00038-5"},{"key":"S002248120000801X_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9754-0_15"},{"key":"S002248120000801X_ref014","doi-asserted-by":"publisher","DOI":"10.1515\/9783110804737"},{"key":"S002248120000801X_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050154"},{"key":"S002248120000801X_ref012","volume-title":"absoluteness in forcing extensions","author":"Schrittesser"},{"key":"S002248120000801X_ref010","unstructured":"Schindler R. , BMM is stronger than BSPFA, preprint, 05 2003."},{"key":"S002248120000801X_ref005","doi-asserted-by":"publisher","DOI":"10.1515\/9783110809114"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000801X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T21:28:13Z","timestamp":1557178093000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000801X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2004,3]]}},"alternative-id":["S002248120000801X"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1080938826","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,3]]}}}