{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T00:12:54Z","timestamp":1782259974938,"version":"3.54.5"},"reference-count":21,"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":3479,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,9]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>A model is said to be <jats:italic>Leibnizian<\/jats:italic> if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom <jats:italic>LM<\/jats:italic> (the Leibniz-Mycielski axiom) such that for any completion <jats:italic>T<\/jats:italic> of Zermelo-Fraenkel set theory <jats:italic>ZF<\/jats:italic>. <jats:italic>T<\/jats:italic> has a Leibnizian model if and only if <jats:italic>T<\/jats:italic> proves <jats:italic>LM<\/jats:italic>. Here we prove:<\/jats:p><jats:p>Theorem A. <jats:italic>Every complete theory T extending ZF + LM has <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007623_inline1\"\/> nonisomorphic countable Leibnizian models<\/jats:italic>.<\/jats:p><jats:p>Theorem B. <jats:italic>If \u03ba is a prescribed definable infinite cardinal ofa complete theory T extending ZF + V = OD, then there are <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007623_inline2\"\/> nonisomorphic Leibnizian models <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007623_inline3\"\/> of T of power<\/jats:italic> \u2135<jats:sub>1<\/jats:sub><jats:italic>such that<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007623_inline4\"\/><jats:italic>is \u2135<jats:sub>1<\/jats:sub>-like<\/jats:italic>.<\/jats:p><jats:p>Theorem C. <jats:italic>Every complete theory T extending<\/jats:italic><jats:bold><jats:italic>ZF<\/jats:italic> + V = OD<\/jats:bold><jats:italic>has <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007623_inline2\"\/> nonisomorphic \u2135<jats:sub>1<\/jats:sub>-like Leibnizian models<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1096901766","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:42:59Z","timestamp":1109799779000},"page":"775-789","source":"Crossref","is-referenced-by-count":7,"title":["Leibnizian models of set theory"],"prefix":"10.1017","volume":"69","author":[{"given":"Ali","family":"Enayat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007623_ref017","first-page":"21","volume-title":"Annals of Mathematical Logic","author":"Mitchell","year":"1972"},{"key":"S0022481200007623_ref011","unstructured":"Enayat A. , Leibnizian models via iterated ultrapowers, in preparation."},{"key":"S0022481200007623_ref001","first-page":"572","volume":"43","author":"Abramson","year":"1978","journal-title":"Models without indiscernibles"},{"key":"S0022481200007623_ref012","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200007623_ref020","volume-title":"Axiomatic set theory, Part 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metamathematics"},{"key":"S0022481200007623_ref019","doi-asserted-by":"publisher","DOI":"10.4064\/fm176-3-1"},{"key":"S0022481200007623_ref021","volume-title":"Proceedings of the Bertrand Russell Memorial Logic Conference, Denmark 1973","author":"Paris","year":"1973"}],"container-title":["Journal of Symbolic 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