{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T05:33:20Z","timestamp":1773898400129,"version":"3.50.1"},"reference-count":13,"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":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:p>In this paper, we shall prove two theorems involving the construction of core models with infinitely many Woodin cardinals. We assume familiarity with [12], which develops core model theory the one Woodin level, and with [10] and [6], which extend the fine structure theory of [5] to mice having many Woodin cardinals. The most important new problem of a general nature which we must face here concerns the iterability of <jats:italic>K<jats:sup>c<\/jats:sup><\/jats:italic> with respect to uncountable iteration trees.<\/jats:p><jats:p>Our first result is the following theorem, a slightly stronger version of which was proved independently and earlier by Woodin. The theorem settles positively a conjecture of Feng, Magidor, and Woodin [2].<\/jats:p><jats:p>Theorem. <jats:italic>Let<\/jats:italic> \u03a9 <jats:italic>be measurable. Then the following are equivalent<\/jats:italic>:<\/jats:p><jats:p>(a) <jats:italic>for all posets<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000952X_inline1\"\/>,<\/jats:p><jats:p>(b) <jats:italic>for every poset<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000952X_inline2\"\/>,<\/jats:p><jats:p>(c) <jats:italic>for every poset<\/jats:italic> \u2119 \u2208 <jats:italic>V<\/jats:italic><jats:sub>\u03a9<\/jats:sub>, <jats:italic>V<\/jats:italic><jats:sup>\u2119<\/jats:sup> \u22a8 <jats:italic>there is no uncountable sequence of distinct reals in L<\/jats:italic>(\u211d)<\/jats:p><jats:p>(d) <jats:italic>there is an \u03a9-iterable premouse of height \u03a9 which satisfies \u201cthere are infinitely many Woodin cardinals\u201d<\/jats:italic>.<\/jats:p><jats:p>It is an immediate corollary that if every set of reals in <jats:italic>L<\/jats:italic>(\u211d) is weakly homogeneous, then AD<jats:sup><jats:italic>L<\/jats:italic>(\u211d)<\/jats:sup> holds. We shall also indicate some extensions of the theorem to pointclasses beyond <jats:italic>L<\/jats:italic>(\u211d), and mice with more than \u03c9 Woodin cardinals.<\/jats:p>","DOI":"10.2178\/jsl\/1190150159","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:14:26Z","timestamp":1197555266000},"page":"1197-1226","source":"Crossref","is-referenced-by-count":20,"title":["Core models with more Woodin cardinals"],"prefix":"10.1017","volume":"67","author":[{"given":"J. R.","family":"Steel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000952X_ref011","first-page":"931","volume":"58","author":"Steel","year":"1993","journal-title":"The wellfoundedness of the Mitchell order"},{"key":"S002248120000952X_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02411-3"},{"key":"S002248120000952X_ref006","first-page":"1285","volume":"64","author":"Neeman","year":"1999","journal-title":"A weak Dodd-Jensen lemma"},{"key":"S002248120000952X_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21903-4"},{"key":"S002248120000952X_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(96)00032-2"},{"key":"S002248120000952X_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1224594-7"},{"key":"S002248120000952X_ref009","unstructured":"Schindler R. D. , Steel J. R. , and Zeman M. , Deconstructing inner model theory, (to appear)."},{"key":"S002248120000952X_ref013","volume-title":"Handbook of set theory","author":"Steel"},{"key":"S002248120000952X_ref002","volume-title":"Universally Baire sets","author":"Feng"},{"key":"S002248120000952X_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90037-E"},{"key":"S002248120000952X_ref007","first-page":"621","volume":"61","author":"Schimmerling","year":"1996","journal-title":"Fine structure for tame inner models"},{"key":"S002248120000952X_ref001","article-title":"The domestic levels of Kc are iterable","author":"Andretta","journal-title":"Israel Journal of Mathematics"},{"key":"S002248120000952X_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22485-4"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000952X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T20:45:56Z","timestamp":1557175556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000952X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S002248120000952X"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150159","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}