{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T12:52:18Z","timestamp":1775479938274,"version":"3.50.1"},"reference-count":0,"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":13798,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,6]]},"abstract":"<jats:p>The extreme interest of set theorists in the notion of \u201cclosed unbounded set\u201d is epitomized in the following well-known theorem:<\/jats:p><jats:p>Theorem A. <jats:italic>For any regular cardinal \u03ba &gt; \u03c9, the intersection of any two closed unbounded subsets of \u03ba is closed and unbounded<\/jats:italic>.<\/jats:p><jats:p>The proof of this theorem is easy and in fact yields a stronger result, namely that for any uncountable regular cardinal \u03ba the intersection of fewer than \u03ba many closed unbounded sets is closed and unbounded. Thus, if, for \u03ba a regular uncountable cardinal, we let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051550_inline1.png\"\/> denote {<jats:italic>A<\/jats:italic> \u2286 \u03ba \u2223 <jats:italic>A<\/jats:italic> contains a closed unbounded subset}, then, for any such \u03ba, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051550_inline1.png\"\/> is a \u03ba-additive nonprincipal filter on \u03ba.<\/jats:p><jats:p>Now what about the possibility of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051550_inline1.png\"\/> being an ultrafilter\u03ba It is routine to see that this is impossible for \u03ba &gt; \u2135<jats:sub>1<\/jats:sub>. However, for \u03ba = \u2135<jats:sub>1<\/jats:sub> the situation is different. If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051550_inline2.png\"\/> were an ultrafilter, \u2135<jats:sub>1<\/jats:sub> would be a measurable cardinal. As is well-known this is impossible if we assume the axiom of choice; however if ZF + \u201cthere exists a measurable cardinal\u201d is consistent, then so is ZF + \u201c\u2135<jats:sub>1<\/jats:sub> is a measurable cardinal\u201d [2]. Furthermore, under the assumption of certain set theoretic axioms (such as the axiom of determinateness or various infinite exponent partition relations) <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051550_inline2.png\"\/> can be proven to be an ultrafilter. (See [3] and [5].)<\/jats:p>","DOI":"10.2307\/2272248","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:41:13Z","timestamp":1146951673000},"page":"481-482","source":"Crossref","is-referenced-by-count":37,"title":["Adding a closed unbounded set"],"prefix":"10.1017","volume":"41","author":[{"given":"J. E.","family":"Baumgartner","sequence":"first","affiliation":[]},{"given":"L. A.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"E. M.","family":"Kleinberg","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051550","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T03:10:20Z","timestamp":1679627420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051550\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,6]]},"references-count":0,"aliases":["10.1017\/s0022481200051550"],"journal-issue":{"issue":"2","published-print":{"date-parts":[[1976,6]]}},"alternative-id":["S0022481200051550"],"URL":"https:\/\/doi.org\/10.2307\/2272248","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1976,6]]}}}