{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T18:48:00Z","timestamp":1767898080978,"version":"3.49.0"},"reference-count":24,"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 either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal \u03ba \u2265 \u2135 such that \u25a1<jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> and \u25a1(<jats:italic>\u03ba<\/jats:italic>) fail. 2) There is a cardinal <jats:italic>\u03ba<\/jats:italic> such that <jats:italic>\u03ba<\/jats:italic> is weakly compact in the generic extension by Col(<jats:italic>\u03ba, \u03ba<\/jats:italic><jats:sup>+<\/jats:sup>). Of special interest is 1) with \u03ba = \u2135<jats:sub>3<\/jats:sub> since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over <jats:italic>K<\/jats:italic><jats:sup>c<\/jats:sup>\u2225<jats:italic>\u03ba<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1231082314","type":"journal-article","created":{"date-parts":[[2009,1,4]],"date-time":"2009-01-04T15:18:44Z","timestamp":1231082324000},"page":"315-335","source":"Crossref","is-referenced-by-count":33,"title":["Stacking mice"],"prefix":"10.1017","volume":"74","author":[{"given":"Ronald","family":"Jensen","sequence":"first","affiliation":[]},{"given":"Ernest","family":"Schimmerling","sequence":"additional","affiliation":[]},{"given":"Ralf","family":"Schindler","sequence":"additional","affiliation":[]},{"given":"John","family":"Steel","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003844_ref024","doi-asserted-by":"publisher","DOI":"10.1515\/9783110857818"},{"key":"S0022481200003844_ref021","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/031\/763902"},{"key":"S0022481200003844_ref019","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22485-4"},{"key":"S0022481200003844_ref018","volume-title":"Handbook of set theory","author":"Steel"},{"key":"S0022481200003844_ref017","unstructured":"Steel J. , The derived model theorem, preprint, math.berkeley.edu\/~steel\/papers\/Publications.html."},{"key":"S0022481200003844_ref016","volume-title":"Handbook of set theory","author":"Schindler"},{"key":"S0022481200003844_ref015","unstructured":"Schindler R. and Steel J. , The core model induction , book in preparation."},{"key":"S0022481200003844_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02411-3"},{"key":"S0022481200003844_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21903-4"},{"key":"S0022481200003844_ref008","first-page":"371","volume":"69","author":"Mitchell","year":"2004","journal-title":"A universal extender model without large cardinals in V"},{"key":"S0022481200003844_ref007","unstructured":"Jensen R. and Steel J. , in preparation."},{"key":"S0022481200003844_ref006","volume-title":"Robust extenders","author":"Jensen","year":"2003"},{"key":"S0022481200003844_ref003","unstructured":"Fuchs G. , Neeman I. , and Schindler R. , A criterion for coarse iterability, Archive for Mathematical Logic , submitted."},{"key":"S0022481200003844_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02773379"},{"key":"S0022481200003844_ref002","volume-title":"Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions","volume":"1476","author":"Bekkali","year":"1991"},{"key":"S0022481200003844_ref005","volume-title":"A new fine structure","author":"Jensen","year":"1997"},{"key":"S0022481200003844_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00036-3"},{"key":"S0022481200003844_ref023","unstructured":"Woodin H. , unpublished."},{"key":"S0022481200003844_ref020","first-page":"1255","volume":"70","author":"Steel","year":"2005","journal-title":"PFA implies ADL(\u211d)"},{"key":"S0022481200003844_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2007.05.005"},{"key":"S0022481200003844_ref022","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(92)90038-M"},{"key":"S0022481200003844_ref004","volume-title":"Set theory","author":"Jech","year":"2002"},{"key":"S0022481200003844_ref010","volume-title":"Handbook of set theory","author":"Schimmerling"},{"key":"S0022481200003844_ref014","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061304000310"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200003844","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T20:00:27Z","timestamp":1556654427000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200003844\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,3]]}},"alternative-id":["S0022481200003844"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1231082314","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,3]]}}}