{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T01:12:47Z","timestamp":1648861967395},"reference-count":8,"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":10511,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,6]]},"abstract":"<jats:p>T. Jech [4] and M. Takahashi [7] proved that given any partial ordering <jats:italic>R<\/jats:italic> in a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200032837_inline1\" \/> of ZFC there is a symmetric submodel of a generic extension of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200032837_inline1\" \/> where <jats:italic>R<\/jats:italic> is isomorphic to the injective ordering on a set of cardinals.<\/jats:p><jats:p>The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of \u201ccardinal universality\u201d is consistent:<\/jats:p><jats:p>CU. <jats:italic>For any partially ordered set (X, \u227c) there is a bijection f:X \u2192 Y such that<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200032837_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>(i.e. <jats:italic>x<\/jats:italic> \u227c <jats:italic>y iff \u2203g: f(x)<\/jats:italic> \u2192 <jats:italic>f(y) injective<\/jats:italic>). (See [1].)<\/jats:p><jats:p>The consistency of CU relative to ZF<jats:sub>0<\/jats:sub> (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.<\/jats:p><jats:p>In this paper the authors define a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200032837_inline2\" \/> of ZF + CU as a symmetric submodel of a generic extension obtained by forcing \u201c\u00e0 la Easton\u201d with a class of conditions which add \u03ba generic subsets to any regular cardinal \u03ba of a ground model satisfying ZF + <jats:italic>V = L<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274238","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:13:30Z","timestamp":1146953610000},"page":"502-509","source":"Crossref","is-referenced-by-count":0,"title":["The consistency of the axiom of universality for the ordering of cardinalities"],"prefix":"10.1017","volume":"50","author":[{"given":"Marco","family":"Forti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Furio","family":"Honsell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200032837_ref004","first-page":"293","article-title":"On ordering of cardinalities","volume":"14","author":"Jech","year":"1966","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences, S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S0022481200032837_ref006","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200032837_ref005","volume-title":"The axiom of choice","author":"Jech","year":"1973"},{"key":"S0022481200032837_ref003","first-page":"493","article-title":"Set theory with free construction principles","volume":"10","author":"Forti","year":"1983","journal-title":"Annali detta Scuola Normale Superiore di Pisa, Classe de Scienze"},{"key":"S0022481200032837_ref001","first-page":"691","volume":"49","author":"Forti","year":"1984","journal-title":"Can cardinal ordering be universal?"},{"key":"S0022481200032837_ref007","first-page":"129","article-title":"On incomparable cardinals","volume":"16","author":"Takahashi","year":"1968","journal-title":"Commentarii Mathematici Universitatis Sancii Pauli"},{"key":"S0022481200032837_ref002","volume-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik","author":"Forti"},{"key":"S0022481200032837_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-27.4.577"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200032837","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T22:36:10Z","timestamp":1558564570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200032837\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1985,6]]}},"alternative-id":["S0022481200032837"],"URL":"https:\/\/doi.org\/10.2307\/2274238","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,6]]}}}