{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T18:00:58Z","timestamp":1649095258732},"reference-count":7,"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":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>In [vDF], van Douwen and Fleissner introduce a number of axioms which hold in models constructed by iteratively forcing MA in a nontrivial extension of the set-theoretic universe. One such model is the Bell-Kunen model, obtained by starting with a model of ZFC + GCH, then forcing \u201cMA + \u03f2 = <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub>\u201d by the standard means, then forcing \u201cMA + \u03f2 = <jats:italic>\u03c9<\/jats:italic><jats:sub>3<\/jats:sub>\u201d, and so on. The Bell-Kunen model is the result of an <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> sequence of extensions of this form, with direct limits taken at limit ordinals. (See [BK] for a more complete description.) Van Douwen and Fleissner observed that many of the properties of this model could be distilled into a \u201cDefinable Forcing Axiom\u201d, which states that \u201cIf <jats:bold>P<\/jats:bold> is a c.c.c. partial order which is definable from a real, then there is a sequence of filters through <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026578_inline2\" \/>, such that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026578_inline1\" \/> is any dense subset of <jats:bold>P<\/jats:bold>, then all but countably many of the \u2131<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>'s meet <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026578_inline1\" \/> in a nonempty set.\u201d (They call such a sequence <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>-generic.) Van Douwen and Fleissner ask whether one can eliminate the restriction on the c.c.c. order entirely; the resulting axiom (\u201cIf <jats:bold>P<\/jats:bold> is any c.c.c. partial order of cardinality at most \u03f2, then there is a sequence of filters\u2026\u201d) is called the <jats:italic>Undefinable Forcing Axiom<\/jats:italic> (UFA).<\/jats:p>","DOI":"10.2307\/2274968","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:33:48Z","timestamp":1146954828000},"page":"284-296","source":"Crossref","is-referenced-by-count":0,"title":["UFA fails in the Bell-Kunen model"],"prefix":"10.1017","volume":"55","author":[{"given":"John W. L.","family":"Merrill","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026578_ref005","volume-title":"Set theory: an introduction to independence proofs","author":"Kunen","year":"1980"},{"key":"S0022481200026578_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-50-2-123-128"},{"key":"S0022481200026578_ref001","first-page":"351","article-title":"On the PI character of ultrafilters","volume":"3","author":"Bell","year":"1981","journal-title":"La Soci\u00e9t\u00e9 Royale du Canada. L'Acad\u00e9mie des Sciences. Comptes Rendues Math\u00e9matiques (Mathematical Reports)"},{"key":"S0022481200026578_ref004","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200026578_ref006","article-title":"A class of consistent anti-Martin's axioms","author":"Merrill","journal-title":"Pacific Journal of Mathematics"},{"key":"S0022481200026578_ref002","unstructured":"van Douwen E. and Fleissner W. , The definable forcing axiom (preprint)."},{"key":"S0022481200026578_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01188048"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026578","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:51:35Z","timestamp":1558216295000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026578\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026578"],"URL":"https:\/\/doi.org\/10.2307\/2274968","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}