{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T11:16:28Z","timestamp":1648725388376},"reference-count":10,"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":12885,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,12]]},"abstract":"<jats:p>An analogue of a theorem of Sierpinski about the classical operation (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049173_inline1\" \/>) provides the motivation for studying <jats:italic>\u03ba<\/jats:italic>-Suslin logic, an extension of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>+<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> which is closed under a propositional connective based on (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049173_inline1\" \/>). This theorem is used to obtain a complete axiomatization for <jats:italic>\u03ba<\/jats:italic>-Suslin logic and an upper bound on the <jats:italic>\u03ba<\/jats:italic>-Suslin accessible ordinals (for <jats:italic>\u03ba<\/jats:italic> = <jats:italic>\u03c9<\/jats:italic> these results are due to Ellentuck [E]). It also yields a weak completeness theorem which we use to generalize a result of Barwise and Kunen [B-K] and show that the least ordinal not <jats:italic>H<\/jats:italic>(<jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>) recursive is the least ordinal not <jats:italic>\u03ba<\/jats:italic>-Suslin accessible.<\/jats:p><jats:p>We assume familiarity with lectures 3, 4 and 10 of Keisler's <jats:bold>Model theory for infinitary logic<\/jats:bold> [Ke]. We use standard notation and terminology including the following.<\/jats:p><jats:p><jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>+<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> is the logic closed under negation, finite quantification, and conjunction and disjunction over sets of formulas of cardinality at most <jats:italic>\u03ba<\/jats:italic>. For <jats:italic>\u03ba<\/jats:italic> singular, conjunctions and disjunctions over sets of cardinality <jats:italic>\u03ba<\/jats:italic> can be replaced by conjunctions and disjunctions over sets of cardinality less than <jats:italic>\u03ba<\/jats:italic> so that we can (and will in \u00a72) assume the formation rules of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>+<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> allow conjunctions and disjunctions only over sets of cardinality strictly less than <jats:italic>\u03ba<\/jats:italic> whenever <jats:italic>\u03ba<\/jats:italic> is singular.<\/jats:p>","DOI":"10.2307\/2273505","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:57Z","timestamp":1146952137000},"page":"659-666","source":"Crossref","is-referenced-by-count":0,"title":["\u03ba-Suslin logic"],"prefix":"10.1017","volume":"43","author":[{"given":"Judy","family":"Green","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049173_ref003","unstructured":"Burgess J.P. , On the Hanf number of Suslin logic (preprint)."},{"key":"S0022481200049173_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-82-2-105-119"},{"key":"S0022481200049173_ref004","first-page":"567","volume":"40","author":"Ellentuck","year":"1975","journal-title":"The foundations of Suslin logic"},{"key":"S0022481200049173_ref007","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200049173_ref006","author":"Green","journal-title":"Some model theory for game logics"},{"key":"S0022481200049173_ref008","volume-title":"Topology","volume":"I","author":"Kurotowski","year":"1966"},{"key":"S0022481200049173_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771648"},{"key":"S0022481200049173_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066791"},{"key":"S0022481200049173_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0081121"},{"key":"S0022481200049173_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049173","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:07:39Z","timestamp":1558984059000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049173\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,12]]},"references-count":10,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1978,12]]}},"alternative-id":["S0022481200049173"],"URL":"https:\/\/doi.org\/10.2307\/2273505","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,12]]}}}