{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T08:52:49Z","timestamp":1698655969392},"reference-count":2,"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 examine the cardinal structure of inner models that satisfy GCH but do not contain 0<jats:sup>#<\/jats:sup>. We show, assuming that 0<jats:sup>#<\/jats:sup> exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal <jats:italic>\u03ba<\/jats:italic> which is <jats:italic>\u03ba<\/jats:italic>-Mahlo. The principal tools are the Covering Theorem for <jats:italic>L<\/jats:italic> and the technique of reverse Easton iteration.<\/jats:p><jats:p>Let <jats:italic>I<\/jats:italic> denote the class of Silver indiscernibles for <jats:italic>L<\/jats:italic> and \u3008<jats:italic>i<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> \u2223 <jats:italic>\u03b1 \u03f5<\/jats:italic> ORD\u3009 its increasing enumeration. Also fix an inner model <jats:italic>M<\/jats:italic> of GCH not containing 0<jats:sup>#<\/jats:sup> and let <jats:italic>\u03c9<\/jats:italic><jats:sub>\u03b1<\/jats:sub> denote the <jats:italic>\u03c9<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> of the model <jats:italic>M<\/jats:italic>[0<jats:sup>#<\/jats:sup>], the least inner model containing <jats:italic>M<\/jats:italic> as a submodel and 0<jats:sup>#<\/jats:sup> as an element.<\/jats:p>","DOI":"10.2178\/jsl\/1190150140","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"924-932","source":"Crossref","is-referenced-by-count":2,"title":["0<sup>#<\/sup> and inner models"],"prefix":"10.1017","volume":"67","author":[{"given":"SY D.","family":"Friedman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009336_ref001","first-page":"115","volume-title":"Marginalia to a theorem of Silver","volume":"499","author":"Devlin","year":"1975"},{"key":"S0022481200009336_ref002","first-page":"158","volume":"63","author":"Friedman","year":"1998","journal-title":"Generic saturation"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009336","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T00:45:35Z","timestamp":1557189935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009336\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S0022481200009336"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150140","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}