{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T03:09:08Z","timestamp":1648609748353},"reference-count":5,"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":12429,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:p>Working in ZFC + Martin's Axiom we develop a generalization of the Barwise Compactness Theorem which holds in languages of cardinality less than <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/>. Next, using this compactness theorem, an omitting types theorem for fewer than <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/> types is proved. Finally, in ZFC, we prove that this compactness result implies Martin's Axiom (the Equivalence Theorem). Our compactness theorem applies to a new class of theories\u2014cc<jats:italic>\u03a3<\/jats:italic>-theories\u2014which generalize the countable <jats:italic>\u03a3<\/jats:italic>-theories of Barwise's theorem. The Omitting Types Theorem and the Equivalence Theorem serve as examples illustrating the use of cc<jats:italic>\u03a3<\/jats:italic>-theories.<\/jats:p><jats:p>Assume <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline2\" \/> = (<jats:italic>A, \u03b5<\/jats:italic>) or <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/> = (<jats:italic>A<\/jats:italic>, <jats:italic>\u03b5 R<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>R<jats:sub>m<\/jats:sub><\/jats:italic>) where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/> is admissible. <jats:italic>L<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/>) is the first-order language with constants for elements of <jats:italic>A<\/jats:italic> and relation symbols for relations in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/>. <jats:italic>L<jats:sub>A<\/jats:sub><\/jats:italic> is <jats:italic>A<\/jats:italic> \u22c2 <jats:italic>L<\/jats:italic><jats:sub>\u221e\u03c9<\/jats:sub> where the <jats:italic>L<\/jats:italic> of <jats:italic>L<\/jats:italic><jats:sub>\u221e\u03c9<\/jats:sub> is any language in <jats:italic>A<\/jats:italic>. A theory <jats:italic>T<\/jats:italic> in <jats:italic>L<jats:sub>A<\/jats:sub><\/jats:italic> is consistent if there is no derivation in <jats:italic>A<\/jats:italic> of a contradiction from <jats:italic>T<\/jats:italic>. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline3\" \/> is <jats:italic>L<jats:sub>A<\/jats:sub><\/jats:italic> with new constants <jats:italic>c<jats:sub>a<\/jats:sub><\/jats:italic> for each <jats:italic>a<\/jats:italic> and <jats:italic>A<\/jats:italic>. The basic terms of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline4\" \/> consist of the constants of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline4\" \/> and the terms <jats:italic>f<\/jats:italic>(<jats:italic>c<jats:sub>a<\/jats:sub><\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>c<jats:sub>a<\/jats:sub><jats:sub>m<\/jats:sub><\/jats:italic>) built directly from constants using functions <jats:italic>f<\/jats:italic> of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline4\" \/>. The symbol <jats:italic>t<\/jats:italic> is used for basic terms. A theory <jats:italic>T<\/jats:italic> in <jats:italic>L<jats:sub>A<\/jats:sub><\/jats:italic> is <jats:italic>\u03a3<\/jats:italic> if it is defined by a <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline5\" \/> formula of <jats:italic>L<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047253_inline1\" \/>). The formula <jats:italic>\u03c6<\/jats:italic><jats:sup>\u231d<\/jats:sup> is a logical equivalent of <jats:italic>\u00ac\u03c6<\/jats:italic> defined by: (1) <jats:italic>\u03c6<\/jats:italic><jats:sup>\u231d<\/jats:sup> = \u00ac<jats:italic>\u03c6<\/jats:italic> if <jats:italic>\u03c6<\/jats:italic> is atomic; (2) (\u00ac<jats:italic>\u03c6<\/jats:italic>)<jats:sup>\u231d<\/jats:sup> = <jats:italic>\u03c6<\/jats:italic> (3) (\u22c1<jats:italic><jats:sub>\u03c6\u2208\u03a6<\/jats:sub> \u03c6<\/jats:italic>)<jats:sup>\u231d<\/jats:sup> = \u22c0<jats:italic><jats:sub>\u03c6\u2208\u03a6<\/jats:sub> \u03c6<\/jats:italic><jats:sup>\u231d<\/jats:sup>; (4) (\u22c0<jats:italic><jats:sub>\u03c6\u2208\u03a6<\/jats:sub> \u03c6<\/jats:italic>) \u22c1<jats:italic><jats:sub>\u03c6\u2208\u03a6<\/jats:sub> \u03c6<\/jats:italic><jats:sup>\u231d<\/jats:sup>; (5) (\u2203<jats:italic>\u03c7\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>))<jats:sup>\u231d<\/jats:sup> \u2200<jats:italic>\u03c7\u03c6<\/jats:italic><jats:sup>\u231d<\/jats:sup>(<jats:italic>x<\/jats:italic>); \u2200<jats:italic>\u03c7\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>))<jats:sup>\u231d<\/jats:sup> = \u2203<jats:italic>\u03c7\u03c6<\/jats:italic><jats:sup>\u231d<\/jats:sup>(<jats:italic>x<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2273364","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:51:56Z","timestamp":1146952316000},"page":"172-176","source":"Crossref","is-referenced-by-count":0,"title":["Martin's axiom in the model theory of <i>L<sub>A<\/sub><\/i>"],"prefix":"10.1017","volume":"45","author":[{"given":"W. Richard","family":"Stark","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200047253_ref005","unstructured":"Stark Richard , A forcing approach to strict-\u03a01 1 reflection and strict-\u03a01 1 = \u03a31 0, with applications in infinitary logic, PH.D. thesis, University of Wisconsin, Madison, 1975."},{"key":"S0022481200047253_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200047253_ref003","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200047253_ref002","first-page":"562","volume":"37","author":"Blass","year":"1972","journal-title":"Theories without countable models"},{"key":"S0022481200047253_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90009-4"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200047253","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T19:36:26Z","timestamp":1558899386000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200047253\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1980,3]]}},"alternative-id":["S0022481200047253"],"URL":"https:\/\/doi.org\/10.2307\/2273364","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,3]]}}}