{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T09:57:45Z","timestamp":1769162265251,"version":"3.49.0"},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2017,2,24]],"date-time":"2017-02-24T00:00:00Z","timestamp":1487894400000},"content-version":"unspecified","delay-in-days":1729,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2012,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at \u2135<jats:sub>2<\/jats:sub> with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's \u2119<jats:sub>max<\/jats:sub> axiom (*) holds, then BPFA implies that <jats:italic>V<\/jats:italic> is closed under the \u201cWoodin-in-the-next-ZFC-model\u201d operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus \u201cNS<jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub> is precipitous\u201d and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].<\/jats:p>","DOI":"10.2178\/jsl\/1333566633","type":"journal-article","created":{"date-parts":[[2012,4,4]],"date-time":"2012-04-04T19:26:53Z","timestamp":1333567613000},"page":"475-498","source":"Crossref","is-referenced-by-count":15,"title":["Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on <i>\u03c9<\/i><sub>1<\/sub>"],"prefix":"10.1017","volume":"77","author":[{"given":"Benjamin","family":"Claverie","sequence":"first","affiliation":[]},{"given":"Ralf","family":"Schindler","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2017,2,24]]},"reference":[{"key":"S0022481200000694_ref024","doi-asserted-by":"publisher","DOI":"10.1007\/BF02780398"},{"key":"S0022481200000694_ref019","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061304000310"},{"key":"S0022481200000694_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9754-0_15"},{"key":"S0022481200000694_ref023","first-page":"751","volume":"74","author":"Schindler","year":"2009","journal-title":"The self-iterability of L[E]"},{"key":"S0022481200000694_ref013","unstructured":"Jensen R. and Steel J. , In Preparation."},{"key":"S0022481200000694_ref002","first-page":"587","article-title":"Saturation properties of ideals in forcing extensions. II","volume":"271","author":"Baumgartner","year":"1982","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200000694_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050154"},{"key":"S0022481200000694_ref031","doi-asserted-by":"publisher","DOI":"10.1515\/9783110804737"},{"key":"S0022481200000694_ref004","first-page":"187","volume":"74","author":"Claverie","year":"2009","journal-title":"Increasing u2 by a stationary set preserving forcing"},{"key":"S0022481200000694_ref008","first-page":"58","volume":"60","author":"Goldstern","year":"1995","journal-title":"The hounded proper forcing axiom"},{"key":"S0022481200000694_ref020","first-page":"1481","volume":"66","author":"Schindler","year":"2001","journal-title":"Proper forcing and remarkable cardinals II"},{"key":"S0022481200000694_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02411-3"},{"key":"S0022481200000694_ref032","doi-asserted-by":"publisher","DOI":"10.1515\/9783110857818"},{"key":"S0022481200000694_ref009","unstructured":"Hjorth G. , The influence of u2 , Ph.D. thesis, UC Berkeley, 1993."},{"key":"S0022481200000694_ref028","unstructured":"Steel J. and Zoble S. , Determinacy from strong reflection, submitted."},{"key":"S0022481200000694_ref030","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-0487-8_8"},{"key":"S0022481200000694_ref014","first-page":"1372","volume":"72","author":"Ketchersid","year":"2007","journal-title":"Increasing \n                     \n                   and Namba style forcing"},{"key":"S0022481200000694_ref022","first-page":"241","volume":"71","author":"Schindler","year":"2006","journal-title":"Iterates of the core model"},{"key":"S0022481200000694_ref015","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.1995.v2.n5.a6"},{"key":"S0022481200000694_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-010-0180-0"},{"key":"S0022481200000694_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2008.12.001"},{"key":"S0022481200000694_ref026","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00021-T"},{"key":"S0022481200000694_ref012","first-page":"315","volume":"74","author":"Jensen","year":"2009","journal-title":"Stacking mice"},{"key":"S0022481200000694_ref016","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00036-3"},{"key":"S0022481200000694_ref027","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-5764-9_20"},{"key":"S0022481200000694_ref025","volume-title":"The core model iterability problem","volume":"8","author":"Steel","year":"1993"},{"key":"S0022481200000694_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/2687750"},{"key":"S0022481200000694_ref021","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200410002"},{"key":"S0022481200000694_ref010","volume-title":"Set theory","author":"Jech","year":"2002"},{"key":"S0022481200000694_ref005","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2009.v16.n5.a4"},{"key":"S0022481200000694_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200000694_ref029","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1090\/conm\/031\/763902","volume-title":"Axiomatic set theory (Boulder, Colorado, 1983)","volume":"31","author":"Todor\u010devi\u0107","year":"1983"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200000694","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T20:47:39Z","timestamp":1556225259000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200000694\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,6]]}},"alternative-id":["S0022481200000694"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1333566633","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,6]]}}}