{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T16:49:54Z","timestamp":1784306994938,"version":"3.55.0"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11059,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/> be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/> has a <jats:italic>\u03a3<jats:sub>n<\/jats:sub><\/jats:italic> end extension for each <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/> is a well-founded model of ZFC, such that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/> has uncountable cofinality and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/> has a topless <jats:italic>\u03a3<\/jats:italic><jats:sub>3<\/jats:sub> end extension, <jats:italic>then<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/> has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200037427_inline1\"\/>) highly hyperinaccessible. In \u00a73 related results are proved for <jats:italic>\u03ba<\/jats:italic>-like models (<jats:italic>\u03ba<\/jats:italic> any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation <jats:italic>\u03ba<\/jats:italic> \u2192 <jats:italic>\u03bb<\/jats:italic>. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a <jats:italic>\u03a3<jats:sub>n<\/jats:sub><\/jats:italic> end extension for each <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>. The case <jats:italic>\u03ba<\/jats:italic> = <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> is investigated more deeply in \u00a74, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>-like models of ZFC are constructed which have a <jats:italic>\u03a3<jats:sub>n<\/jats:sub><\/jats:italic> end extension for all <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic> but have no elementary end extension. <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>-like models of ZFC which have no <jats:italic>\u03a3<\/jats:italic><jats:sub>3<\/jats:sub> end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for <jats:italic>\u03ba<\/jats:italic>-like models with <jats:italic>\u03ba<\/jats:italic> regular, topless end extensions are much rarer than blunt end extensions.<\/jats:p>","DOI":"10.2307\/2273669","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:06:05Z","timestamp":1146953165000},"page":"1053-1073","source":"Crossref","is-referenced-by-count":7,"title":["Blunt and topless end extensions of models of set theory"],"prefix":"10.1017","volume":"48","author":[{"given":"Matt","family":"Kaufmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200037427_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200037427_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771605"},{"key":"S0022481200037427_ref005","first-page":"331","volume-title":"Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics","author":"Keisler","year":"1974"},{"key":"S0022481200037427_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-71-1-43-62"},{"key":"S0022481200037427_ref014","first-page":"531","volume":"37","author":"Schmerl","year":"1972","journal-title":"On power-like models for hyperinaccessible cardinals"},{"key":"S0022481200037427_ref011","unstructured":"Ratajczyk Z. , On axiomatization of ZF KM and ZF KM (to appear)."},{"key":"S0022481200037427_ref016","first-page":"821","volume-title":"Handbook of Mathematical Logic","author":"Smory\u0144ski","year":"1978"},{"key":"S0022481200037427_ref003","first-page":"139","volume":"41","author":"Hutchinson","year":"1976","journal-title":"Elementary extensions of countable models of set theory"},{"key":"S0022481200037427_ref007","first-page":"177","volume-title":"Axiomatic Set Theory (Part 1), Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Keisler","year":"1970"},{"key":"S0022481200037427_ref004","first-page":"183","article-title":"Blunt and topless end extensions","volume":"3","author":"Kaufmann","year":"1982","journal-title":"Abstracts of the American Mathemetical Society"},{"key":"S0022481200037427_ref008","first-page":"257","volume-title":"Infinitistic Methods, Proceedings of Symposium in Foundations of Mathematics (Warsaw, 1959)","author":"MacDowell","year":"1961"},{"key":"S0022481200037427_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-2-219-242"},{"key":"S0022481200037427_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-36-1-143-164"},{"key":"S0022481200037427_ref012","first-page":"521","volume":"37","author":"Schmerl","year":"1972","journal-title":"An elementary sentence which has ordered models"},{"key":"S0022481200037427_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90011-5"},{"key":"S0022481200037427_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90009-8"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200037427","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T22:11:56Z","timestamp":1558649516000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200037427\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,12]]},"references-count":16,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1983,12]]}},"alternative-id":["S0022481200037427"],"URL":"https:\/\/doi.org\/10.2307\/2273669","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,12]]}}}