{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T02:16:33Z","timestamp":1772504193131,"version":"3.50.1"},"reference-count":9,"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":1837,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2009,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that if <jats:italic>I<\/jats:italic> is a precipitous ideal on <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> and if <jats:italic>\u03b8<\/jats:italic> &gt; <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> is a regular cardinal, then there is a forcing \u2119 = \u2119(<jats:italic>I, \u03b8<\/jats:italic>) which preserves the stationarity of all <jats:italic>I<\/jats:italic>-positive sets such that in <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup>, \u27e8<jats:italic>H<jats:sub>\u03b8<\/jats:sub><\/jats:italic>; \u2208, <jats:italic>I<\/jats:italic>\u27e9 is a generic iterate of a countable structure \u27e8<jats:italic>M<\/jats:italic>; \u2208, <jats:italic>\u012a<\/jats:italic>\u27e9. This shows that if the nonstationary ideal on \u03c9<jats:sub>1<\/jats:sub> is precipitous and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003789_inline4\"\/> exists, then there is a stationary set preserving forcing which increases <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003789_inline1\"\/>. Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> is precipitous, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003789_inline2\"\/>.<\/jats:p>","DOI":"10.2178\/jsl\/1231082308","type":"journal-article","created":{"date-parts":[[2009,1,4]],"date-time":"2009-01-04T15:18:44Z","timestamp":1231082324000},"page":"187-200","source":"Crossref","is-referenced-by-count":3,"title":["Increasing <i>u<\/i><sub>2<\/sub> by a stationary set preserving forcing"],"prefix":"10.1017","volume":"74","author":[{"given":"Benjamin","family":"Claverie","sequence":"first","affiliation":[]},{"given":"Ralf","family":"Schindler","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003789_ref006","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200003789_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1971415"},{"key":"S0022481200003789_ref002","volume-title":"Set theory","author":"Jech","year":"2003"},{"key":"S0022481200003789_ref003","volume-title":"Making cardinals \u03c9-cofinal","author":"Jensen","year":"1990"},{"key":"S0022481200003789_ref004","volume-title":"On some problems of Mitchell, Welch and Vickers","author":"Jensen","year":"1990"},{"key":"S0022481200003789_ref005","first-page":"1372","volume":"72","author":"Ketchersid","year":"2007","journal-title":"Increasing  and Namba style forcing"},{"key":"S0022481200003789_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200410002"},{"key":"S0022481200003789_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1982-0656481-5"},{"key":"S0022481200003789_ref009","doi-asserted-by":"publisher","DOI":"10.1515\/9783110804737"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200003789","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T20:00:50Z","timestamp":1556654450000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200003789\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,3]]}},"alternative-id":["S0022481200003789"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1231082308","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,3]]}}}