{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T08:19:26Z","timestamp":1648628366223},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12520,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,12]]},"abstract":"<jats:p>Solovay proved in 1967 that the axiom of determinateness implies that the filter <jats:italic>C<\/jats:italic> generated by closed and unbounded subsets of \u03c9<jats:sub>1<\/jats:sub> is an ultrafilter. It has long been conjectured that a significant part of the theory of the axiom of determinateness should be provable from the hypothesis that <jats:italic>C<\/jats:italic> is an ultrafilter, but even the first step of finding inner models with several measurable cardinals has proved elusive. In this paper we show that such models exist. Much of our proof is a modification of Kunen's proof in [3] of the same conclusion from the existence of a measurable cardinal \u03ba such that 2<jats:sup>\u03ba<\/jats:sup> &gt; \u03ba<jats:sup>+<\/jats:sup>.<\/jats:p><jats:p>Since no proof of Solovay's result seems to have been published, we insert a proof here. We want to show that for any set <jats:italic>x<\/jats:italic> \u2282 \u03c9<jats:sub>1<\/jats:sub> there is a closed, unbounded set either contained in or disjoint from <jats:italic>x<\/jats:italic>. By the lemma of [4] there is a Turing degree d such that either \u03c9<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">e<\/jats:sup> \u0404 <jats:italic>x<\/jats:italic> for all degrees e \u2265<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub> d or \u03c9<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">e<\/jats:sup> \u2209 <jats:italic>x<\/jats:italic> for all degrees e \u2265<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub> d. By a theorem of Sacks [1], [5] every d-admissible is \u03c9<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">e<\/jats:sup> for some e \u2265<jats:sup><jats:italic>T<\/jats:italic><\/jats:sup> d, so it is enough to show that there is a closed, unbounded set of d-admissibles. Let <jats:italic>a<\/jats:italic> \u2282 \u03c9 have degree d; then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048386_inline1\" \/> is such a set.<\/jats:p>","DOI":"10.2307\/2273289","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:51:21Z","timestamp":1146952281000},"page":"503-506","source":"Crossref","is-referenced-by-count":0,"title":["On the ultrafilter of closed, unbounded sets"],"prefix":"10.1017","volume":"44","author":[{"given":"D. A.","family":"Martin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"W.","family":"Mitchell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048386_ref001","first-page":"77","volume-title":"Lecture Notes in Mathematics","author":"Friedman","year":"1968"},{"key":"S0022481200048386_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1968-11995-0"},{"key":"S0022481200048386_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90013-6"},{"key":"S0022481200048386_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(76)90187-0"},{"key":"S0022481200048386_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71228-X"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048386","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T20:28:00Z","timestamp":1558902480000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048386\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,12]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1979,12]]}},"alternative-id":["S0022481200048386"],"URL":"https:\/\/doi.org\/10.2307\/2273289","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,12]]}}}