{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:54:27Z","timestamp":1775462067542,"version":"3.50.1"},"reference-count":6,"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":15167,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,9]]},"abstract":"<jats:p>Let &lt; and \u227c be two distinguished binary relation symbols. A structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120007907X_inline1\"\/> is <jats:italic>\u03ba-like<\/jats:italic> iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120007907X_inline2\"\/> is a linear ordering of <jats:italic>A<\/jats:italic>, card(<jats:italic>A<\/jats:italic>) = <jats:italic>\u03ba<\/jats:italic>, and every proper initial segment of <jats:italic>A<\/jats:italic> has cardinality &lt; <jats:italic>\u03ba<\/jats:italic>. A structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120007907X_inline1\"\/> is <jats:italic>\u03b1-ordered<\/jats:italic> iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120007907X_inline3\"\/> is a (reflexive) linear ordering of type <jats:italic>\u03b1<\/jats:italic> with field a subset of <jats:italic>A<\/jats:italic>. We define when a cardinal <jats:italic>\u03ba<\/jats:italic> is <jats:italic>\u03b1-inaccessible<\/jats:italic>. (In this paper, inaccessible always means weakly inaccessible.) The 0-inaccessible cardinals are just the inaccessible cardinals; if <jats:italic>\u03b1<\/jats:italic> &gt; 0, then <jats:italic>\u03ba<\/jats:italic> is <jats:italic>\u03b1<\/jats:italic>-inaccessible iff for each <jats:italic>\u03b2 &lt; \u03b1<\/jats:italic>, each closed, cofinal subset of <jats:italic>\u03ba<\/jats:italic> contains a <jats:italic>\u03b2<\/jats:italic>-inaccessible. (The (1 + <jats:italic>\u03b1<\/jats:italic>)-inaccessibles are just the <jats:italic>\u03c1<jats:sub>\u03b1<\/jats:sub><\/jats:italic> cardinals of Mahlo.) This paper is concerned with the proof of the following theorem.<\/jats:p><jats:p>Main Theorem. <jats:italic>There is an elementary sentence \u03c3 with the property that whenever \u03b1 is an ordinal and \u03ba an infinite cardinal, then \u03c3 has an \u03b1-ordered \u03ba-like model iff \u03ba is not \u03b1-inaccessible<\/jats:italic>.<\/jats:p><jats:p>This theorem gives some additional answers to a question of Mostowski about languages with generalized quantifiers. Fuhrken [1] showed that this question is equivalent to the following one: <jats:italic>For which cardinals \u03ba and \u03bb is it true that if an elementary sentence has a \u03ba-like model, then it has a \u03bb-like model?<\/jats:italic> It is actually this question to which the theorem refers. The theorem limits the possible pairs <jats:italic>\u03ba, \u03bb<\/jats:italic> of cardinals which answer the question. In fact, if the question is generalized so as to permit sentences from some more extensive language, then the theorem still limits the possible answers. For a more thorough introduction to this problem, the reader is referred to the aforementioned article of Fuhrken as well as Keisler [2] and Vaught [6].<\/jats:p>","DOI":"10.2307\/2272738","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:17:55Z","timestamp":1146950275000},"page":"521-530","source":"Crossref","is-referenced-by-count":5,"title":["An elementary sentence which has ordered models"],"prefix":"10.1017","volume":"37","author":[{"given":"James H.","family":"Schmerl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120007907X_ref003","first-page":"843","article-title":"On hyperaccessible-like models","volume":"16","author":"Schmerl","year":"1969","journal-title":"Notices of the American Mathematical Society"},{"key":"S002248120007907X_ref005","first-page":"531","volume":"37","author":"Schmerl","year":"1972","journal-title":"On power-like models for hyperinaccessible cardinals"},{"key":"S002248120007907X_ref001","first-page":"121","volume-title":"The theory of models","author":"Fuhrken","year":"1965"},{"key":"S002248120007907X_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71186-8"},{"key":"S002248120007907X_ref004","unstructured":"Schmerl J. H. , On \u03ba-like models for inaccessible \u03ba, Doctoral Dissertation, University of California, Berkeley, 1971."},{"key":"S002248120007907X_ref006","first-page":"81","volume-title":"Logic, methodology and philosophy of science","author":"Vaught","year":"1965"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120007907X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T20:57:48Z","timestamp":1559249868000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120007907X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1972,9]]}},"alternative-id":["S002248120007907X"],"URL":"https:\/\/doi.org\/10.2307\/2272738","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,9]]}}}