{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T06:12:08Z","timestamp":1648879928565},"reference-count":13,"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":5398,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let \u03ba<jats:sup>\u211d<\/jats:sup> be the least ordinal \u03ba such that <jats:italic>L<\/jats:italic><jats:sub>\u03ba<\/jats:sub> (\u211d) is admissible. Let <jats:italic>A<\/jats:italic> = {<jats:italic>x<\/jats:italic> \u03f5 \u211d \u2223 (\u2203\u03b1 &lt; \u03ba<jats:sup>\u211d<\/jats:sup>) such that <jats:italic>x<\/jats:italic> is ordinal definable in <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub> (\u211d)}. It is well known that (assuming determinacy) <jats:italic>A<\/jats:italic> is the largest countable inductive set of reals. Let <jats:italic>T<\/jats:italic> be the theory: ZFC \u2212 Replacement + \u201cThere exists \u03c9 Woodin cardinals which are cofinal in the ordinals.\u201d <jats:italic>T<\/jats:italic> has consistency strength weaker than that of the theory ZFC + \u201cThere exists \u03c9 Woodin cardinals\u201d, but stronger than that of the theory ZFC + \u201cThere exists <jats:italic>n<\/jats:italic> Woodin Cardinals\u201d, for each <jats:italic>n<\/jats:italic> \u03f5 \u03c9. Let <jats:italic>M<\/jats:italic> be the canonical, minimal inner model for the theory <jats:italic>T<\/jats:italic>. In this paper we show that <jats:italic>A<\/jats:italic> = \u211d \u2229 <jats:italic>M<\/jats:italic>. Since <jats:italic>M<\/jats:italic> is a <jats:italic>mouse<\/jats:italic>, we say that <jats:italic>A<\/jats:italic> is a <jats:italic>mouse set<\/jats:italic>. As an application, we use our characterization of <jats:italic>A<\/jats:italic> to give an inner-model-theoretic proof of a theorem of Martin which states that for all <jats:italic>n<\/jats:italic>, every <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200013396_inline1\" \/> real is in <jats:italic>A<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2586477","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:03:01Z","timestamp":1146938581000},"page":"443-459","source":"Crossref","is-referenced-by-count":3,"title":["The largest countable inductive set is a mouse set"],"prefix":"10.1017","volume":"64","author":[{"given":"Mitch","family":"Rudominer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013396_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21903-4"},{"key":"S0022481200013396_ref004","volume-title":"Set theory: An introduction to independence proofs","volume":"102","author":"Kunen","year":"1980"},{"key":"S0022481200013396_ref006","first-page":"97\u2013106","volume-title":"Cabal seminar 79\u201381","volume":"1019","author":"Martin","year":"1983"},{"key":"S0022481200013396_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90037-E"},{"key":"S0022481200013396_ref001","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200013396_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0419235-7"},{"key":"S0022481200013396_ref005","unstructured":"Martin D. A. , untitled book on large cardinals and determinacy, In preparation."},{"key":"S0022481200013396_ref010","first-page":"10\u2013156","volume-title":"Cabal seminar 79\u201381","volume":"1019","author":"Steel","year":"1983"},{"key":"S0022481200013396_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0071702"},{"key":"S0022481200013396_ref013","unstructured":"Steel J. R. , A theorem of Woodin's on mouse sets, Unpublished notes, 1996."},{"key":"S0022481200013396_ref008","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200013396_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)89645-5"},{"key":"S0022481200013396_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00021-T"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013396","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T21:18:11Z","timestamp":1557436691000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013396\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,6]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1999,6]]}},"alternative-id":["S0022481200013396"],"URL":"https:\/\/doi.org\/10.2307\/2586477","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,6]]}}}