{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T04:21:54Z","timestamp":1768278114275,"version":"3.49.0"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":17451,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1966,6]]},"abstract":"<jats:p>We study certain first and second order theories which are semantically defined as the sets of all sentences true in certain given structures. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200067360_inline1\"\/> be a <jats:italic>structure<\/jats:italic> where <jats:italic>A<\/jats:italic> is a non-empty set, \u03bb is an ordinal, and <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> is an <jats:italic>n<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>)-ary relation or function<jats:sup>4<\/jats:sup> on <jats:italic>A<\/jats:italic>. With <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200067360_inline2\"\/> we associate a language <jats:italic>L<\/jats:italic> appropriate for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200067360_inline2\"\/> which may be a first or higher order calculus. <jats:italic>L<\/jats:italic> has an <jats:italic>n<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>)-place predicate or function constant <jats:bold>P<\/jats:bold> for each <jats:italic>\u03b1<\/jats:italic> &lt; \u03bb. We shall study three types of languages: (1) first-order calculi with equality; (2) second-order monadic calculi which contain monadic predicate (set) variables ranging over subsets of <jats:italic>A<\/jats:italic>; (3) restricted (weak) second-order calculi which contain monadic predicate variables ranging over <jats:italic>finite<\/jats:italic> subsets of <jats:italic>A<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2269808","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T20:33:08Z","timestamp":1146947588000},"page":"169-181","source":"Crossref","is-referenced-by-count":92,"title":["Decidability and undecidability of extensions of second (first) order theory of (generalized) successor"],"prefix":"10.1017","volume":"31","author":[{"given":"Calvin C.","family":"Elgot","sequence":"first","affiliation":[]},{"given":"Michael O.","family":"Rabin","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200067360_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1958-0093479-4"},{"key":"S0022481200067360_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1950-0038912-1"},{"key":"S0022481200067360_ref006","doi-asserted-by":"publisher","DOI":"10.1147\/rd.32.0114"},{"key":"S0022481200067360_ref001","first-page":"834","volume":"5","year":"1959","journal-title":"American Mathematical Society Notices"},{"key":"S0022481200067360_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1961-0139530-9"},{"key":"S0022481200067360_ref003","first-page":"1","journal-title":"Logic Methodology and Philosophy of Sciences, Proceedings of the I960 International Congress"},{"key":"S0022481200067360_ref002","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19600060105"},{"key":"S0022481200067360_ref005","first-page":"105","volume":"11","year":"1946","journal-title":"this Journal"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200067360","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,2]],"date-time":"2019-06-02T20:59:36Z","timestamp":1559509176000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200067360\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1966,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1966,6]]}},"alternative-id":["S0022481200067360"],"URL":"https:\/\/doi.org\/10.2307\/2269808","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1966,6]]}}}