{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T18:35:37Z","timestamp":1759775737865},"reference-count":12,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T00:00:00Z","timestamp":1490918400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/100003187","name":"National Sleep Foundation","doi-asserted-by":"publisher","award":["DMS\u20100800762"],"award-info":[{"award-number":["DMS\u20100800762"]}],"id":[{"id":"10.13039\/100003187","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["209252"],"award-info":[{"award-number":["209252"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2017,4]]},"abstract":"<jats:p>We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of \u03c9<jats:sub>1<\/jats:sub>\u2010like models of set theory. Specifically, under the \u22c4 hypothesis and suitable consistency assumptions, we show that there is a family of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201500002-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201500002:malq201500002-math-0001\" \/> many \u03c9<jats:sub>1<\/jats:sub>\u2010like models of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201500002-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201500002:malq201500002-math-0002\" \/>, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive \u03c9<jats:sub>1<\/jats:sub>\u2010like model of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201500002-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201500002:malq201500002-math-0003\" \/> that does not embed into its own constructible universe; and there can be an \u03c9<jats:sub>1<\/jats:sub>\u2010like model of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201500002-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201500002:malq201500002-math-0004\" \/> whose structure of hereditarily finite sets is not universal for the \u03c9<jats:sub>1<\/jats:sub>\u2010like models of set theory.<\/jats:p>","DOI":"10.1002\/malq.201500002","type":"journal-article","created":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T15:12:33Z","timestamp":1490973153000},"page":"66-76","source":"Crossref","is-referenced-by-count":1,"title":["Incomparable \u03c9<sub>1<\/sub>\u2010like models of set theory"],"prefix":"10.1002","volume":"63","author":[{"given":"Gunter","family":"Fuchs","sequence":"first","affiliation":[{"name":"Department of Mathematics The Graduate Center of The City University of New York 365 Fifth Avenue New York NY 10016 United States of America"},{"name":"Department of Mathematics College of Staten Island, City University of New York 2800 Victory Boulevard Staten Island NY 10314 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Victoria","family":"Gitman","sequence":"additional","affiliation":[{"name":"Department of Mathematics The Graduate Center of The City University of New York 365 Fifth Avenue New York NY 10016 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joel David","family":"Hamkins","sequence":"additional","affiliation":[{"name":"Department of Mathematics The Graduate Center of The City University of New York 365 Fifth Avenue New York NY 10016 United States of America"},{"name":"Department of Mathematics College of Staten Island, City University of New York 2800 Victory Boulevard Staten Island NY 10314 United States of America"},{"name":"Department of Philosophy New York University 5 Washington Place New York NY 10003 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2017,3,31]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1984-0737894-1"},{"key":"e_1_2_7_3_1","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/BFb0066789","volume-title":"Proceedings of the Cambridge Summer School in Mathematical Logic, held in Cambridge, UK, 1\u201321 August 1971","author":"Friedman H.","year":"1973"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061313500062"},{"issue":"2","key":"e_1_2_7_5_1","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1090\/S0002-9939-1977-0476498-7","article-title":"A rather classless model","volume":"62","author":"Kaufmann M.","year":"1977","journal-title":"Proc. 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