{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T12:07:22Z","timestamp":1754482042619},"reference-count":6,"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":3845,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2003,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Tarski [5] showed that for any set<jats:italic>X<\/jats:italic>, its set<jats:italic>\u03c9<\/jats:italic>(<jats:italic>X<\/jats:italic>) of well-orderable subsets has cardinality strictly greater than that of<jats:italic>X<\/jats:italic>, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation \u2223<jats:italic>\u03c9<\/jats:italic>(<jats:italic>X<\/jats:italic>)\u2223 = \u2223<jats:italic>Y<\/jats:italic>\u2223. This contrasts with the corresponding situation for power sets, where use of Hartogs' \u2135-function easily establishes that there can be no infinite descending sequence under the relation<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200008598_inline1\" \/>.<\/jats:p>","DOI":"10.2178\/jsl\/1058448446","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:10:06Z","timestamp":1109797806000},"page":"879-884","source":"Crossref","is-referenced-by-count":3,"title":["Non-well-foundedness of well-orderable power sets"],"prefix":"10.1017","volume":"68","author":[{"given":"T. E.","family":"Forster","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. K.","family":"Truss","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200008598_ref003","first-page":"774","article-title":"A refinement of Cantor's theorem","volume":"83","author":"Kirmayer","year":"1981","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200008598_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm-32-1-176-783"},{"key":"S0022481200008598_ref002","first-page":"351","article-title":"Applications of the \u03b8-model","volume":"14","author":"Jech","year":"1966","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences"},{"key":"S0022481200008598_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(00)00031-2"},{"key":"S0022481200008598_ref006","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1093\/oso\/9780198534686.003.0005","volume-title":"Automorphisms of first-order structures","author":"Truss","year":"1994"},{"key":"S0022481200008598_ref001","first-page":"323","volume":"56","author":"Forster","year":"1991","journal-title":"End-extensions preserving power set"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200008598","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T22:45:43Z","timestamp":1706049943000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200008598\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2003,9]]}},"alternative-id":["S0022481200008598"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1058448446","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,9]]}}}