{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T14:01:11Z","timestamp":1775829671485,"version":"3.50.1"},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14621,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,3]]},"abstract":"<jats:p>The cardinality problem for ultraproducts is as follows: Given an ultrafilter <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> over a set <jats:italic>I<\/jats:italic> and cardinals \u03b1<jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>i<\/jats:italic> \u2208 <jats:italic>I<\/jats:italic>, what is the cardinality of the ultraproduct <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline2\"\/>? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> over a set <jats:italic>I<\/jats:italic> of power <jats:italic>\u03ba<\/jats:italic> (uniform means that all elements of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> have power <jats:italic>\u03ba<\/jats:italic>). It is open whether every countably incomplete <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> has the property that, for all infinite \u03b1, the ultra-power <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline3\"\/> has power <jats:italic>\u03b1<jats:sup>\u03ba<\/jats:sup><\/jats:italic>. However, it is shown in [4] that certain countably incomplete <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/>, namely the <jats:italic>\u03ba<\/jats:italic>-regular <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/>, have this property.<\/jats:p><jats:p>This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the <jats:italic>\u03ba<\/jats:italic>-good ultrafilters, do have this property. The <jats:italic>\u03ba<\/jats:italic>-good ultrafilters are important in model theory because they are exactly the ultrafilters <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> such that every ultraproduct modulo <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> is <jats:italic>\u03ba<\/jats:italic>-saturated (see [5]).<\/jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"\/> be an ultrafilter on a set <jats:italic>I<\/jats:italic>. Let \u03b1<jats:sub><jats:italic>i, n<\/jats:italic><\/jats:sub>, <jats:italic>i<\/jats:italic> \u2208 <jats:italic>I<\/jats:italic>, <jats:italic>n<\/jats:italic> \u2208 \u03c9, be cardinals and \u03b1<jats:sub><jats:italic>i, n<\/jats:italic><\/jats:sub>, \u2265 \u03b1<jats:sub><jats:italic>i, m<\/jats:italic><\/jats:sub> if <jats:italic>n<\/jats:italic> &lt; <jats:italic>m<\/jats:italic>. Let<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_eqnU1\"\/><\/jats:disp-formula>.<\/jats:p><jats:p>Then <jats:italic>\u03c1<jats:sub>n<\/jats:sub><\/jats:italic> are nonincreasing and therefore there is some <jats:italic>m<\/jats:italic> and <jats:italic>\u03c1<\/jats:italic> such that <jats:italic>\u03c1<jats:sub>n<\/jats:sub><\/jats:italic> = <jats:italic>\u03c1<\/jats:italic> if <jats:italic>n<\/jats:italic> \u2265 <jats:italic>m<\/jats:italic>. We call <jats:italic>\u03c1<\/jats:italic> the <jats:italic>eventual value<\/jats:italic> (abbreviated ev val) of <jats:italic>\u03c1<jats:sub>n<\/jats:sub><\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2272341","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:27:55Z","timestamp":1146950875000},"page":"43-48","source":"Crossref","is-referenced-by-count":3,"title":["A result concerning cardinalities of ultraproducts"],"prefix":"10.1017","volume":"39","author":[{"given":"H. Jerome","family":"Keisler","sequence":"first","affiliation":[]},{"given":"Karel","family":"Prikry","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200065403_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/1970386"},{"key":"S0022481200065403_ref001","unstructured":"Eklof P. C. , The structure of uhraproducts of Abelian groups (to appear)."},{"key":"S0022481200065403_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1972-0314619-7"},{"key":"S0022481200065403_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90012-5"},{"key":"S0022481200065403_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/1970549"},{"key":"S0022481200065403_ref005","first-page":"112","volume-title":"Proceedings of the 1964 International Congress","author":"Keisler","year":"1965"},{"key":"S0022481200065403_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11653-7"},{"key":"S0022481200065403_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1964-11219-2"},{"key":"S0022481200065403_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-51-3-195-228"},{"key":"S0022481200065403_ref007","first-page":"47","volume":"32","author":"Keisler","year":"1967","journal-title":"Ultraproducts of finite sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200065403","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T21:37:26Z","timestamp":1559165846000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200065403\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1974,3]]}},"alternative-id":["S0022481200065403"],"URL":"https:\/\/doi.org\/10.2307\/2272341","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,3]]}}}