{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T23:25:22Z","timestamp":1767914722336,"version":"3.49.0"},"reference-count":11,"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":2933,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2006,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let<jats:italic>N<\/jats:italic>be a transitive model of ZFC such that \u201c<jats:italic>N<\/jats:italic>\u2282<jats:italic>N<\/jats:italic>and<jats:italic>P<\/jats:italic>(\u211d) \u2282<jats:italic>N<\/jats:italic>. Assume that both<jats:italic>V<\/jats:italic>and<jats:italic>N<\/jats:italic>satisfy \u201cthe core model<jats:italic>K<\/jats:italic>exists.\u201d Then<jats:italic>K<\/jats:italic><jats:sup>N<\/jats:sup>is an iterate of<jats:italic>K<\/jats:italic>, i.e., there exists an iteration tree<jats:italic>F<\/jats:italic>on<jats:italic>K<\/jats:italic>such that<jats:italic>F<\/jats:italic>has successor length and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006447_inline1\"\/>. Moreover, if there exists an elementary embedding \u03c0:<jats:italic>V<\/jats:italic>\u2192<jats:italic>N<\/jats:italic>then the iteration map associated to the main branch of<jats:italic>F<\/jats:italic>equals \u03c0 \u0575<jats:italic>K<\/jats:italic>. (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that<jats:italic>P<\/jats:italic>(\u211d) \u2282<jats:italic>N<\/jats:italic>is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.<\/jats:p>","DOI":"10.2178\/jsl\/1140641172","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:22:31Z","timestamp":1198081351000},"page":"241-251","source":"Crossref","is-referenced-by-count":15,"title":["Iterates of the core model"],"prefix":"10.1017","volume":"71","author":[{"given":"Ralf","family":"Schindler","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200006447_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600586"},{"key":"S0022481200006447_ref002","volume-title":"The core model for non-overlapping extender sequences","author":"Jensen","year":"1991"},{"key":"S0022481200006447_ref009","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1201\/9781439865736-15","volume-title":"Logic Colloquium '01","volume":"20","author":"Schindler","year":"2005"},{"key":"S0022481200006447_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1224594-7"},{"key":"S0022481200006447_ref003","unstructured":"Jensen R. , On some problems of Mitchell, Welch, and Vickers, handwritten notes."},{"key":"S0022481200006447_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(96)00032-2"},{"key":"S0022481200006447_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21903-4"},{"key":"S0022481200006447_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00113-0"},{"key":"S0022481200006447_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22485-4"},{"key":"S0022481200006447_ref011","first-page":"1197","volume":"67","author":"Steel","year":"2002","journal-title":"Core models with more Woodin cardinals"},{"key":"S0022481200006447_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00103-8"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200006447","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T00:53:50Z","timestamp":1630371230000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200006447\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,3]]}},"alternative-id":["S0022481200006447"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1140641172","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,3]]}}}