{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T18:31:25Z","timestamp":1649097085108},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14621,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,3]]},"abstract":"<jats:p>Let \u03c3 be any sequence <jats:italic>B<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub> \u2026, <jats:italic>B<jats:sub>n<\/jats:sub><\/jats:italic>, \u2026 of transitive sets closed under pairs with <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120006549X_inline1\" \/> for each <jats:italic>n<\/jats:italic>. In this paper we show that the smallest admissible set <jats:italic>A<jats:sub>\u03c3<\/jats:sub><\/jats:italic> with <jats:italic>\u03c3<\/jats:italic> \u2208 <jats:italic>A<jats:sub>\u03c3<\/jats:sub><\/jats:italic> is \u03a3<jats:sub>1<\/jats:sub> compact. Thus we have an entirely new class of explicitly describable uncountable \u03a3<jats:sub>1<\/jats:sub> compact sets.<\/jats:p><jats:p>The search for uncountable \u03a3<jats:sub>1<\/jats:sub> compact languages goes back to Hanf's negative results on compact cardinals [7]. Barwise first showed that all countable admissible sets were \u03a3<jats:sub>1<\/jats:sub> compact [1] and then went on to give a characterization of the \u03a3<jats:sub>1<\/jats:sub> compact sets in terms of strict <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120006549X_inline2\" \/> reflection [2]. While his characterization has been of interest in understanding the \u03a3<jats:sub>1<\/jats:sub> compactness phenomenon it has led to the identification of only one class of uncountable \u03a3<jats:sub>1<\/jats:sub> compact sets. In particular, Barwise showed [2], using the above notation, that if \u22c3<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:italic>B<\/jats:italic><jats:sub>n<\/jats:sub> is power set admissible it satisfies the strict <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120006549X_inline2\" \/> reflection principle and hence is \u03a3<jats:sub>1<\/jats:sub> compact. (This result was obtained independently by Karp using algebraic methods [9].)<\/jats:p><jats:p>In proving our compactness theorem we follow Makkai's approach to the Barwise Compactness Theorem [12] and use a modified version of Smullyan's abstract consistency property [14]. A direct generalization of Makkai's method to the cofinality \u03c9 case yields a proof of the Barwise-Karp result mentioned above [6]. In order to obtain our new result we depart from the usual definition of language and use instead the indexed languages of Karp [9] in which a conjunction is considered to operate on a function whose range is a set of formulas rather than on a set of formulas itself.<\/jats:p>","DOI":"10.2307\/2272350","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:27:55Z","timestamp":1146936475000},"page":"105-116","source":"Crossref","is-referenced-by-count":6,"title":["\u03a3<sub>1<\/sub> compactness for next admissible sets"],"prefix":"10.1017","volume":"39","author":[{"given":"Judy","family":"Green","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120006549X_ref014","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.49.6.828"},{"key":"S002248120006549X_ref013","unstructured":"Platek R. , Foundations of recursion theory, Ph.D. Thesis, Stanford University, Stanford, Calif., 1966."},{"key":"S002248120006549X_ref012","first-page":"341","article-title":"An application of a method of Smullyan to logics on admissible sets","volume":"17","author":"Makkai","year":"1969","journal-title":"Bulletin de l'Academie Polonaise des Sciences. Serie des Sciences Mathematiques, Astronomiques et Physiques"},{"key":"S002248120006549X_ref009","first-page":"80","volume-title":"Lecture Notes in Mathematics","author":"Karp","year":"1968"},{"key":"S002248120006549X_ref011","volume-title":"Memoirs of the American Mathematical Society","author":"L\u00e9vy","year":"1965"},{"key":"S002248120006549X_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-53-3-309-324"},{"key":"S002248120006549X_ref003","first-page":"1","volume-title":"Lecture Notes in Mathematics","author":"Barwise","year":"1968"},{"key":"S002248120006549X_ref006","unstructured":"Green J. , Consistency properties for uncountable finite-quantifier languages, Ph.D. Thesis, University of Maryland, College Park, Md., 1972."},{"key":"S002248120006549X_ref004","first-page":"108","volume":"36","author":"Barwise","year":"1971","journal-title":"The next admissible set"},{"key":"S002248120006549X_ref001","first-page":"226","volume":"34","author":"Barwise","year":"1969","journal-title":"Infinitary logic and admissible sets"},{"key":"S002248120006549X_ref002","first-page":"409","volume":"34","author":"Barwise","year":"1969","journal-title":"Applications of strict  predicates to infinitary logic"},{"key":"S002248120006549X_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771648"},{"key":"S002248120006549X_ref008","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Jensen","year":"1971"},{"key":"S002248120006549X_ref010","volume-title":"Model theory for infinitary logic","author":"Keisler","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\/S002248120006549X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:37:58Z","timestamp":1559151478000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120006549X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1974,3]]}},"alternative-id":["S002248120006549X"],"URL":"https:\/\/doi.org\/10.2307\/2272350","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,3]]}}}