{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T23:12:49Z","timestamp":1649027569890},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13433,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1977,6]]},"abstract":"<jats:p>In this note we give a simple recursive axiomatization for the class of structures of type (\u2136<jats:sub>\u03c9<\/jats:sub> \u2135<jats:sub>0<\/jats:sub>). This solves a problem of Vaught which is Problem 13 in the book [1] of Chang and Keisler. The same technique is used to get a recursive axiomatization for the class of \u03ba-like structures where \u03ba is strongly \u03c9-inaccessible.<\/jats:p><jats:p>Let us fix throughout some recursive first-order language <jats:italic>L<\/jats:italic>, and until further notice let us suppose that included in <jats:italic>L<\/jats:italic> is a distinguished unary predicate symbol <jats:italic>U<\/jats:italic>. For cardinals \u03ba and \u03bb with \u03ba \u2265 \u03bb \u2265 \u2135<jats:sub>0<\/jats:sub>, we say the structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200050337_inline01\" \/> has type (\u03ba, \u03bb) if card(<jats:italic>A<\/jats:italic>)= \u03ba and card <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200050337_inline02\" \/>. Let <jats:italic>K<\/jats:italic>(\u03ba, \u03bb) be the class of all structures of type (\u03ba, \u03bb). For each ordinal \u03b1 define 2<jats:sub arrange=\"stack\">\u03b1<\/jats:sub><jats:sup arrange=\"stack\">\u03ba<\/jats:sup>by 2<jats:sub arrange=\"stack\">0<\/jats:sub><jats:sup arrange=\"stack\">\u03ba<\/jats:sup> = \u03ba, and 2<jats:sub arrange=\"stack\">\u03b1<\/jats:sub><jats:sup arrange=\"stack\">\u03ba<\/jats:sup>= \u22c3 {2<jats:sup>\u03bb<\/jats:sup>: \u03bb = 2<jats:sub arrange=\"stack\">\u03b2<\/jats:sub><jats:sup arrange=\"stack\">\u03ba<\/jats:sup> for some \u03b2 &lt; \u03b1} when \u03b1 &gt; 0. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200050337_inline03\" \/><\/jats:p><jats:p>Vaught proved the following theorem in [7].<\/jats:p><jats:p>Theorem (Vaught). <jats:italic>Suppose a is a sentence such that for each n &lt; \u03c9 there are<\/jats:italic> \u03ba, \u03bb <jats:italic>with<\/jats:italic> \u03ba &gt; 2<jats:sub arrange=\"stack\">\u03bb<\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sup> and a model of \u03c3 of type (\u03ba, \u03bb). <jats:italic>Then whenever<\/jats:italic> \u03ba \u2265 \u03bb \u2265 \u2135<jats:sub>0<\/jats:sub>, <jats:italic>the sentence \u03c3 has a model of type<\/jats:italic> (\u03ba, \u03bb).<\/jats:p>","DOI":"10.2307\/2272117","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:44:08Z","timestamp":1146951848000},"page":"174-178","source":"Crossref","is-referenced-by-count":3,"title":["An axiomatization for a class of two-cardinal models"],"prefix":"10.1017","volume":"42","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":"S0022481200050337_ref006","first-page":"531","volume":"37","author":"Schmerl","year":"1972","journal-title":"On power-like models for hyperinaccessible cardinals"},{"key":"S0022481200050337_ref001","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200050337_ref005","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10648"},{"key":"S0022481200050337_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1966-11456-8"},{"key":"S0022481200050337_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771640"},{"key":"S0022481200050337_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1956-10036-0"},{"key":"S0022481200050337_ref007","first-page":"390","volume-title":"The theory of models","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\/S0022481200050337","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T21:35:52Z","timestamp":1558992952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200050337\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1977,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1977,6]]}},"alternative-id":["S0022481200050337"],"URL":"https:\/\/doi.org\/10.2307\/2272117","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1977,6]]}}}