{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T13:30:27Z","timestamp":1648819827443},"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":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>One of the main results of Barwise [2] (see also [7, Chapter VIII]) showed that the <jats:italic>s<\/jats:italic> \u2013 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline4\" \/> reflection principle for a set <jats:italic>A<\/jats:italic> is equivalent to \u03a3<jats:sub>1<\/jats:sub>-compactness of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline1\" \/>. Here <jats:italic>A<\/jats:italic> is any transitive p.r. closed set, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline1\" \/> is the infinitary language on <jats:italic>A<\/jats:italic> which allows conjunction and disjunction over arbitrary sets \u03a6 \u0404 <jats:italic>A<\/jats:italic>, and finite quantification.<\/jats:p><jats:p>In this paper we consider languages <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline2\" \/>, where <jats:italic>B<\/jats:italic> is a \u0394<jats:sub>0<\/jats:sub> subset of <jats:italic>A<\/jats:italic>, which is like <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline1\" \/> but we allow quantifiers \u2200<jats:italic>x<\/jats:italic> and \u2203<jats:italic>x<\/jats:italic> where <jats:italic>x<\/jats:italic> is any set of variables indexed by an element of <jats:italic>B<\/jats:italic>. A treatment similar to that of [2] for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline1\" \/> establishes a sufficient, and in some cases necessary, condition for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline2\" \/> to be \u03a3<jats:sub>1<\/jats:sub>-compact. The use of infinitary Skolem functions is intrinsic to the method, so to avoid a separate development of the rudiments of the Skolem language <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline3\" \/> we actually define <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049409_inline2\" \/> to have <jats:italic>b<\/jats:italic>-ary relation and function symbols for every <jats:italic>b<\/jats:italic> \u0404 <jats:italic>B<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273528","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:21Z","timestamp":1146952101000},"page":"508-520","source":"Crossref","is-referenced-by-count":1,"title":["\u03a3<sub>1<\/sub>-compactness in languages stronger than"],"prefix":"10.1017","volume":"43","author":[{"given":"Nigel","family":"Cutland","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049409_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200049409_ref006","unstructured":"Stark W. R. , Compactness and completeness for large languages (to appear)."},{"key":"S0022481200049409_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0281602"},{"key":"S0022481200049409_ref003","first-page":"108","volume":"36","author":"Barwise","year":"1971","journal-title":"The next admissible set"},{"key":"S0022481200049409_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079680"},{"key":"S0022481200049409_ref002","first-page":"409","volume":"34","author":"Barwise","year":"1969","journal-title":"Applications of strict \u03a011-predicates to infinitary logic"},{"key":"S0022481200049409_ref004","first-page":"1","volume-title":"Axiomatic set theory, Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Chang","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\/S0022481200049409","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:33:43Z","timestamp":1558985623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049409\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049409"],"URL":"https:\/\/doi.org\/10.2307\/2273528","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}