{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T11:26:22Z","timestamp":1773401182431,"version":"3.50.1"},"reference-count":12,"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":2749,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2006,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper investigates when it is possible for a partial ordering \u2119 to force <jats:italic>P<jats:sub>k<\/jats:sub>(\u039b)\\V<\/jats:italic> to be stationary in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup>. It follows from a result of Gitik that whenever \u2119 adds a new real, then <jats:italic>P<jats:sub>k<\/jats:sub>(\u039b)\\V<\/jats:italic> is stationary in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup> for each regular uncountable cardinal \u03ba in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup> and all cardinals \u03bb \u2265 \u03ba in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup> [4], However, a covering theorem of Magidor implies that when no new \u03c9-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of \u03c9<jats:sub>1<\/jats:sub>-Erd\u0151s cardinals: If \u2119 is \u2135<jats:sub>1<\/jats:sub>-Cohen forcing, then <jats:italic>P<jats:sub>k<\/jats:sub>(\u039b)\\V<\/jats:italic> is stationary in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup>, for all regular \u03ba \u2265 \u2135<jats:sub>2<\/jats:sub>and all \u03bb \u2269 \u03ba. The following is equiconsistent with an \u03c9<jats:sup>1<\/jats:sup>-Erd\u0151s cardinal: If \u2119 is \u2135<jats:sub>1<\/jats:sub>-Cohen forcing, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006046_inline1\"\/> is stationary in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup>. The following is equiconsistent with \u03ba measurable cardinals: If \u2119 is \u03ba-Cohen forcing, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006046_inline2\"\/> is stationary in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup>.<\/jats:p>","DOI":"10.2178\/jsl\/1154698589","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:23:20Z","timestamp":1198081400000},"page":"1029-1043","source":"Crossref","is-referenced-by-count":4,"title":["Co-stationarity of the ground model"],"prefix":"10.1017","volume":"71","author":[{"given":"Natasha","family":"Dobrinen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sy-David","family":"Friedman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200006046_ref010","first-page":"1","article-title":"Independence proof of (\u03c91, \u03c9\u03b1)-distributive law in complete Boolean algebras","volume":"19","author":"Namba","year":"1971","journal-title":"Commentarii Mathematici Universitatis Sancti Pauli"},{"key":"S0022481200006046_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0939805-5"},{"key":"S0022481200006046_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90026-7"},{"key":"S0022481200006046_ref004","first-page":"881","volume":"50","author":"Gitik","year":"1985","journal-title":"Nonsplitting subsets of"},{"key":"S0022481200006046_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90006-N"},{"key":"S0022481200006046_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(91)90047-P"},{"key":"S0022481200006046_ref011","first-page":"505","volume":"45","author":"Shelah","year":"1980","journal-title":"Independence of strong partition relation for small cardinals, and the free-subset problem"},{"key":"S0022481200006046_ref006","volume-title":"Hanbook of Boolean algebra","volume":"1","author":"Koppelberg","year":"1989"},{"key":"S0022481200006046_ref002","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511629198"},{"key":"S0022481200006046_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12831-2"},{"key":"S0022481200006046_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(75)90009-1"},{"key":"S0022481200006046_ref009","volume-title":"Handbook of set theory","author":"Mitchell"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200006046","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,2]],"date-time":"2019-05-02T19:54:06Z","timestamp":1556826846000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200006046\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,9]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2006,9]]}},"alternative-id":["S0022481200006046"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1154698589","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,9]]}}}