{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T21:32:03Z","timestamp":1697664723707},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that a form of the Erd\u04e7s property (consistent with <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>[H\u03c9<jats:sub>2<\/jats:sub>] and strictly weaker than the Weak Chang's Conjecture at \u03c9<jats:sub>1<\/jats:sub>), together with Bounded Martin's Maximum implies that Woodin's principle <jats:italic>\u03c8<\/jats:italic><jats:sub><jats:italic>AC<\/jats:italic><\/jats:sub> holds, and therefore <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009476_inline1\" \/>. We also prove that <jats:italic>\u03c8<\/jats:italic><jats:sub><jats:italic>AC<\/jats:italic><\/jats:sub> implies that every function <jats:italic>f<\/jats:italic>: \u03c9<jats:sub>1<\/jats:sub> \u2192 \u03c9<jats:sub>1<\/jats:sub> is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.<\/jats:p>","DOI":"10.2178\/jsl\/1190150154","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"1141-1152","source":"Crossref","is-referenced-by-count":7,"title":["Bounded Martin's Maximum, weak Erd\u04e7s cardinals, and <i>\u03c8<\/i><sub><i>Ac<\/i><\/sub>"],"prefix":"10.1017","volume":"67","author":[{"given":"David","family":"Asper\u00f3","sequence":"first","affiliation":[]},{"given":"Philip D.","family":"Welch","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009476_ref016","volume-title":"Series in logic and its applications, number 1","author":"Woodin","year":"1999"},{"key":"S0022481200009476_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/1971415"},{"key":"S0022481200009476_ref007","first-page":"233","article-title":"On the consistency strength of \u2018accessible\u2019 Jo\u00b4nsson cardinals","volume":"25","author":"Donder","year":"1983","journal-title":"Annals of Mathematical Logic"},{"key":"S0022481200009476_ref006","unstructured":"Deiser Oliver and Donder Hans-Dieter , Canonical Functions, non-regular ultrafilters and Ulam's problem on \u03c91 , submitted."},{"key":"S0022481200009476_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90012-8"},{"key":"S0022481200009476_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(00)00058-0"},{"key":"S0022481200009476_ref015","unstructured":"Welch Philip , On unfoldable cardinals, \u03c9-closed cardinals, and the beginning of the inner model hierarchy, submitted."},{"key":"S0022481200009476_ref009","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610798006462"},{"key":"S0022481200009476_ref013","doi-asserted-by":"publisher","DOI":"10.1142\/9789812812940_0011"},{"key":"S0022481200009476_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050154"},{"key":"S0022481200009476_ref011","first-page":"58","volume":"60","author":"Goldstern","year":"1995","journal-title":"The bounded proper forcing axiom"},{"key":"S0022481200009476_ref001","unstructured":"Asper\u00f3 David , Boundedforcing axioms and the continuum, Ph.D. thesis, University of Barcelona, 2000."},{"key":"S0022481200009476_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90030-4"},{"key":"S0022481200009476_ref002","unstructured":"Asper\u00f3 David , Bounded forcing axioms and the size of the continuum, submitted."},{"key":"S0022481200009476_ref014","first-page":"360","volume":"52","author":"Shelah","year":"1987","journal-title":"Semiproperforcing axiom implies Martin maximum but not PFA+"},{"key":"S0022481200009476_ref012","volume-title":"Perspectives in mathematical logic","author":"Jech","year":"1997"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009476","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T00:45:26Z","timestamp":1557189926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009476\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S0022481200009476"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150154","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}