{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T09:09:01Z","timestamp":1773392941701,"version":"3.50.1"},"reference-count":7,"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":14802,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1973,9]]},"abstract":"<jats:p>Each of the various \u201clarge cardinal\u201d axioms currently studied in set theory owes its inspiration to concrete phenomena in various fields. For example, the statement of the well-known compactness theorem for first-order logic can be generalized in various ways to infinitary languages to yield definitions of compact cardinals, and the reflection principles provable in ZF, when modified in the appropriate way, yields indescribable cardinals.<\/jats:p><jats:p>In this paper we concern ourselves with two kinds of large cardinals which are probably the two best known of those whose origins lie in model theory. They are the Rowbottom cardinals and the Jonsson cardinals.<\/jats:p><jats:p>Let us be more specific. A cardinal <jats:italic>\u03ba<\/jats:italic> is said to be a Jonsson cardinal if every structure of cardinality <jats:italic>\u03ba<\/jats:italic> has a proper elementary substructure of cardinality <jats:italic>\u03ba<\/jats:italic>. (It is routine to see that only uncountable cardinals can be Jonsson. Erd\u00f6s and Hajnal have shown [2] that for <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic> no \u2135<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is Jonsson. (In fact, they showed that if <jats:italic>\u03ba<\/jats:italic> is not Jonsson then neither is the successor cardinal of <jats:italic>\u03ba<\/jats:italic> and that, assuming GCH, no successor cardinal can be Jonsson.) Keisler and Rowbottom first showed that the existence of a Jonsson cardinal contradicts <jats:italic>V = L<\/jats:italic>.) The definition of a Rowbottom cardinal is only slightly more intricate. We assume for the moment that our similarity type has a designated one-place relation.<\/jats:p>","DOI":"10.2307\/2273038","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:24:44Z","timestamp":1146950684000},"page":"423-427","source":"Crossref","is-referenced-by-count":10,"title":["Rowbottom cardinals and Jonsson cardinals are almost the same"],"prefix":"10.1017","volume":"38","author":[{"given":"E. M.","family":"Kleinberg","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054785_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90009-X"},{"key":"S0022481200054785_ref001","unstructured":"Devlin K. , Some weak versions of large cardinal axioms, Doctoral Dissertation, University of Bristol, 1971."},{"key":"S0022481200054785_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90010-6"},{"key":"S0022481200054785_ref002","first-page":"19","article-title":"On a problem of B. Jonsson","volume":"14","author":"Erd\u00f6s","year":"1966","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences. S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S0022481200054785_ref003","first-page":"827","article-title":"Rowbottom cardinals and Jonsson cardinals are almost the same","volume":"18","author":"Kleinberg","year":"1971","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200054785_ref004","unstructured":"Kleinberg E. M. , The equiconsistency of two large cardinal axioms (to appear)."},{"key":"S0022481200054785_ref005","volume-title":"The abstract compactness idea","author":"Kleinberg","year":"1971"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054785","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T19:38:06Z","timestamp":1559245086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054785\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1973,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1973,9]]}},"alternative-id":["S0022481200054785"],"URL":"https:\/\/doi.org\/10.2307\/2273038","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1973,9]]}}}