{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:39:40Z","timestamp":1775644780508,"version":"3.50.1"},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6493,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,6]]},"abstract":"<jats:p>In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017461_inline1\"\/> models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].<\/jats:p><jats:p><jats:italic>Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017461_inline2\"\/> such that<\/jats:italic><\/jats:p><jats:p>(1) <jats:italic>every level of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017461_inline1\"\/> is a sound, tame mouse, and<\/jats:italic><\/jats:p><jats:p>(2) <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017461_inline1\"\/> \u22a8 \u201c<jats:italic>There is a strong cardinal that is a limit of Woodin cardinals\u201d<\/jats:italic>.<\/jats:p><jats:p>Recall that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017461_inline1\"\/> satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].<\/jats:p><jats:p><jats:italic>Suppose that fi is a normal measure on<\/jats:italic> \u03a9 <jats:italic>and that all premice are tame. Then K<jats:sup>c<\/jats:sup>, the background certified core model, exists and is a premouse of height<\/jats:italic> \u03a9. <jats:italic>Moreover<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017461_inline3\"\/><jats:italic>for \u03bc-almost every \u03b1<\/jats:italic> &lt; \u03a9.<\/jats:p><jats:p>Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.<\/jats:p><jats:p><jats:italic>If there is a cardinal \u03ba such that \u03ba is \u03ba<jats:sup>+<\/jats:sup>-strongly compact, then there is a premouse that is not tame<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275679","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:58:29Z","timestamp":1146956309000},"page":"621-639","source":"Crossref","is-referenced-by-count":13,"title":["Fine structure for tame inner models"],"prefix":"10.1017","volume":"61","author":[{"given":"E.","family":"Schimmerling","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. R.","family":"Steel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017461_ref004","volume-title":"The core model iterability problem","author":"Steel"},{"key":"S0022481200017461_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1224594-7"},{"key":"S0022481200017461_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00036-3"},{"key":"S0022481200017461_ref005","unstructured":"Steel J. R. , Core models with more Woodin cardinals, preprint."},{"key":"S0022481200017461_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21903-4"},{"key":"S0022481200017461_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90037-E"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017461","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T21:28:36Z","timestamp":1557696516000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017461\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1996,6]]}},"alternative-id":["S0022481200017461"],"URL":"https:\/\/doi.org\/10.2307\/2275679","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,6]]}}}