{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T17:16:44Z","timestamp":1673630204041},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>. We shall show, among other things, that if <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic> then for each cardinal \u03bb there is a set <jats:italic>A<\/jats:italic> \u2286 \u03bb such that neither <jats:italic>A<\/jats:italic> nor \u03bb \u2013 <jats:italic>A<\/jats:italic> contain a closed set of type \u03c9<jats:sub>1<\/jats:sub>. This is an extension of a result of Silver who proved it for \u03bb = \u03c9<jats:sub>2<\/jats:sub>, providing a partial answer to Problem 68 of Friedman [2].<\/jats:p><jats:p>The main results of this paper were obtained independently by both authors.<\/jats:p><jats:p>If \u03bb is an ordinal, <jats:italic>E<\/jats:italic> is said to be Mahlo (or stationary) in \u03bb, if \u03bb \u2013 <jats:italic>E<\/jats:italic> does not contain a closed cofinal subset of \u03bb.<\/jats:p><jats:p>Consider the statements:<\/jats:p><jats:p>(J<jats:sub>1<\/jats:sub>) There is a class <jats:italic>E<\/jats:italic> of limit ordinals and a sequence <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub> defined on singular limit ordinals \u03bb such that<\/jats:p><jats:p>(i) <jats:italic>E<\/jats:italic> \u22c2 \u03bc is Mahlo in \u03bc for all regular &gt; \u03c9;<\/jats:p><jats:p>(ii) <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub> is closed and unbounded in \u03bb;<\/jats:p><jats:p>(iii) if \u03b3 &lt; is a limit point of <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub>, then \u03b3 is singular, \u03b3 \u2209 <jats:italic>E<\/jats:italic> and <jats:italic>C<\/jats:italic><jats:sub>\u03b3<\/jats:sub> = \u03b3 \u22c2 <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub>.<\/jats:p><jats:p>For each infinite cardinal \u03ba:<\/jats:p><jats:p>(J<jats:sub>2,\u03ba<\/jats:sub>) There is a set <jats:italic>E<\/jats:italic> \u2282 \u03ba<jats:sup>+<\/jats:sup> and a sequence <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub>(Lim(\u03bb), \u03bb &lt; \u03ba<jats:sup>+<\/jats:sup>) such that<\/jats:p><jats:p>(i) <jats:italic>E<\/jats:italic> is Mahlo in \u03ba<jats:sup>+<\/jats:sup>;<\/jats:p><jats:p>(ii) <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub> is closed and unbounded in \u03bb;<\/jats:p><jats:p>(iii) if cf(\u03bb) &lt; \u03ba, then card <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub> &lt; \u03ba;<\/jats:p><jats:p>(iv) if \u03b3 &lt; \u03bb is a limit point of <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub> then \u03b3 \u2209 <jats:italic>E<\/jats:italic> and <jats:italic>C<\/jats:italic><jats:sub>\u03b3<\/jats:sub> = \u03e3 \u22c2 <jats:italic>C<\/jats:italic><jats:sub>\u03bb<\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2272274","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:33:55Z","timestamp":1146951235000},"page":"75-80","source":"Crossref","is-referenced-by-count":27,"title":["On partitions into stationary sets"],"prefix":"10.1017","volume":"40","author":[{"given":"Karel","family":"Prikry","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert M.","family":"Solovay","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120005430X_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90014-4"},{"key":"S002248120005430X_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1974-0327521-9"},{"key":"S002248120005430X_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S002248120005430X_ref002","unstructured":"Friedman H. , Ninety-four problems in mathematical logic (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\/S002248120005430X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T20:06:08Z","timestamp":1559160368000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120005430X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S002248120005430X"],"URL":"https:\/\/doi.org\/10.2307\/2272274","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}