{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T08:28:17Z","timestamp":1649147297593},"reference-count":5,"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":15076,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,12]]},"abstract":"<jats:p>This paper is devoted to a description of the way in which ultraproducts can be used in proofs of various well-known \u03a3<jats:sub>1<\/jats:sub>-compactness theorems for infinitary languages <jats:italic>\u2112<jats:sub>A<\/jats:sub><\/jats:italic> associated with admissible sets <jats:italic>A<\/jats:italic>; the method generalises the ultra-product proof of compactness for finitary languages.<\/jats:p><jats:p>The compactness theorems we consider are (\u00a72) the Barwise Compactness Theorem for <jats:italic>\u2112<jats:sub>A<\/jats:sub><\/jats:italic> when <jats:italic>A<\/jats:italic> is countable admissible [1], and (\u00a73) the Cofinality (<jats:italic>\u03c9<\/jats:italic>) Compactness Theorem of Barwise and Karp [2] and [4]. Our proof of the Barwise theorem unfortunately has the defect that it relies heavily on the Completeness Theorem for <jats:italic>\u2112<jats:sub>A<\/jats:sub><\/jats:italic>. This defect has, however, been avoided in the case of the Cf(<jats:italic>\u03c9<\/jats:italic>) Compactness Theorem, so we have a purely model-theoretic proof.<\/jats:p>","DOI":"10.2307\/2272411","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:19:43Z","timestamp":1146935983000},"page":"668-672","source":"Crossref","is-referenced-by-count":0,"title":["\u03a3<sub>1<\/sub>-compactness and ultraproducts"],"prefix":"10.1017","volume":"37","author":[{"given":"Nigel","family":"Cutland","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200078464_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079684"},{"key":"S0022481200078464_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0281602"},{"key":"S0022481200078464_ref001","first-page":"226","volume":"34","author":"Barwise","year":"1969","journal-title":"Infinitary logic and admissible sets"},{"key":"S0022481200078464_ref005","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200078464_ref002","first-page":"409","volume":"34","author":"Barwise","year":"1969","journal-title":"Strict  predicates"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200078464","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T16:32:27Z","timestamp":1559233947000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200078464\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,12]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1972,12]]}},"alternative-id":["S0022481200078464"],"URL":"https:\/\/doi.org\/10.2307\/2272411","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,12]]}}}