{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T11:29:31Z","timestamp":1772623771900,"version":"3.50.1"},"reference-count":6,"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":6128,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1997,6]]},"abstract":"<jats:p>Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol \u2192 and the other is the metalogical symbol \u21d2 in sequents<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200016339_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>Considering the logical system LJ as a mathematical object, we understand that the logical symbols \u2227, \u2228, \u2192, \u00ac, \u2200, \u2203 are operators on formulas, and \u21d2 is a relation. That is, <jats:italic>\u03c6<\/jats:italic> \u21d2 <jats:italic>\u03a8<\/jats:italic> is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which <jats:italic>\u03c6<\/jats:italic> \u21d2 <jats:italic>\u03a8<\/jats:italic> is a sentence.<\/jats:p><jats:p>The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.<\/jats:p><jats:p>Let \u03a9 be a cHa. \u03a9 can be a truth value set of a model of LJ. Then the logical symbols \u2227, \u2228, \u2192, \u00ac, \u2200<jats:italic>x<\/jats:italic>, \u2203<jats:italic>x<\/jats:italic> are interpreted as operators on \u03a9, and the sentence <jats:italic>\u03c6<\/jats:italic> \u21d2 <jats:italic>\u03a8<\/jats:italic> is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol \u21d2 also can be expressed as a logical operators such that <jats:italic>\u03c6<\/jats:italic> \u21d2 <jats:italic>\u03a8<\/jats:italic> is interpreted as 1 or 0.<\/jats:p>","DOI":"10.2307\/2275544","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T23:01:20Z","timestamp":1146956480000},"page":"506-528","source":"Crossref","is-referenced-by-count":9,"title":["Completeness of global intuitionistic set theory"],"prefix":"10.1017","volume":"62","author":[{"given":"Satoko","family":"Titani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200016339_ref005","first-page":"291","volume-title":"The mathematics of fuzzy systems","author":"Takeuti","year":"1985"},{"key":"S0022481200016339_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061825"},{"key":"S0022481200016339_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90081-9"},{"key":"S0022481200016339_ref002","first-page":"221","volume":"40","author":"Powell","year":"1975","journal-title":"Extending G\u00f6del's negative interpretation of ZF"},{"key":"S0022481200016339_ref003","unstructured":"Rasiowa H. and Sikorski R. , Mathematics of metamathematics, 1963."},{"key":"S0022481200016339_ref004","volume-title":"Proof theory","author":"Takeuti","year":"1987"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200016339","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T20:55:55Z","timestamp":1557608155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200016339\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1997,6]]}},"alternative-id":["S0022481200016339"],"URL":"https:\/\/doi.org\/10.2307\/2275544","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,6]]}}}