{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T16:13:17Z","timestamp":1775664797695,"version":"3.50.1"},"reference-count":7,"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":12703,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,6]]},"abstract":"<jats:p>Large cardinal properties divide rather strikingly into two groups. \u201cSmall\u201d large cardinal properties such as weak compactness always relativize to <jats:italic>L<\/jats:italic>, while in contrast \u201clarge\u201d large cardinal properties such as Ramsey are incompatible with <jats:italic>L<\/jats:italic>. These properties seem to be similar otherwise and this sense of similarity is reinforced by the fact that many of the large cardinals do exist in the <jats:italic>L<\/jats:italic>-like model <jats:italic>L(\u03bc)<\/jats:italic>. This paper will show that the division is caused by an artificially restrictive class of \u201cconstructible\u201d sets rather than an essential difference in the properties themselves. Specifically, we consider the class <jats:italic>K<\/jats:italic> of sets constructible from mice as defined by Dodd and Jensen [3] and prove<\/jats:p><jats:p>Theorem 1. <jats:italic>If \u03c1 is Ramsey then \u03c1 is Ramsey in K<\/jats:italic>.<\/jats:p><jats:p>A modification of the proof will show<\/jats:p><jats:p>Theorem 2. <jats:italic>If \u03c1 is Jonson, then \u03c1 is Ramsey in K<\/jats:italic>.<\/jats:p><jats:p>Since Ramsey implies Jonson this shows that the notions are equiconsistent. Theorem 2 was proved by Kunen [5] under the assumption that <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>(\u03bc).<\/jats:p><jats:p>The proof of Theorems 1 and 2 depends heavily on results of Dodd and Jensen [3] about <jats:italic>K<\/jats:italic> and mice. These results are stated without proof in \u00a72. \u00a72 also contains elementary (to a reader familiar with the theory of iterated ultrapowers) proofs of special cases of some of these lemmas sufficient to give a self contained proof of the following corollary of Theorem 1:<\/jats:p><jats:p>Corollary. <jats:italic>If \u03c1 \u2264 \u03ba,\u03c1 is Ramsey and L(\u03bc) \u22a8 \u03bc is a measure on \u03ba then L(\u03bc) \u22a8 \u03c1 is Ramsey<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273732","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:50:09Z","timestamp":1146952209000},"page":"260-266","source":"Crossref","is-referenced-by-count":18,"title":["Ramsey cardinals and constructibility"],"prefix":"10.1017","volume":"44","author":[{"given":"William","family":"Mitchell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048969_ref006","unstructured":"Mitchell W. , Constructibility and large cardinals (in preparation)."},{"key":"S0022481200048969_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90013-6"},{"key":"S0022481200048969_ref004","volume-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik","author":"Henle"},{"key":"S0022481200048969_ref003","unstructured":"Dodd T. and Jensen R. , The core model, preprint, 1976."},{"key":"S0022481200048969_ref002","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-25.4.249"},{"key":"S0022481200048969_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02023868"},{"key":"S0022481200048969_ref007","unstructured":"Silver J. , Large cardinals and the continuum hypothesis (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048969","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T21:20:27Z","timestamp":1558905627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048969\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1979,6]]}},"alternative-id":["S0022481200048969"],"URL":"https:\/\/doi.org\/10.2307\/2273732","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,6]]}}}