{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:55:01Z","timestamp":1759146901187},"reference-count":26,"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":7589,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Working within weak subsystems of second-order arithmetic <jats:bold>Z<\/jats:bold><jats:sub>2<\/jats:sub> we consider two versions of the Baire Category theorem which are not equivalent over the base system RCA<jats:sub>0<\/jats:sub>. We show that one version (B.C.T.I) is provable in RCA<jats:sub>0<\/jats:sub> while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of <jats:bold>Z<\/jats:bold><jats:sub>2<\/jats:sub>, which we call and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200021320_inline1\" \/>, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200021320_inline2\" \/>, show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200021320_inline1\" \/> suffices to prove B.C.T.II. Some model theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200021320_inline2\" \/> and its importance in view of Hilbert's program is discussed, as well as applications of our results to functional analysis.<\/jats:p>","DOI":"10.2307\/2275219","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:48:26Z","timestamp":1146941306000},"page":"557-578","source":"Crossref","is-referenced-by-count":18,"title":["The Baire category theorem in weak subsystems of second-order arithmetic"],"prefix":"10.1017","volume":"58","author":[{"given":"Douglas K.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Stephen G.","family":"Simpson","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200021320_ref022","first-page":"783","volume":"49","author":"Simpson","year":"1984","journal-title":"Which set existence axioms are needed to prove the Cauchy-Peano theorem for ordinary differential equations?"},{"key":"S0022481200021320_ref014","volume-title":"Real analysis","author":"Royden","year":"1968"},{"key":"S0022481200021320_ref009","unstructured":"Hirst J. , Combinatorics in subsystems of second order arithmetic, Ph.D. Thesis , The Pennsylvania State University, University Park, PA, 1987."},{"key":"S0022481200021320_ref015","first-page":"117","volume-title":"Algebras of sets binumerable in complete extensions of arithmetic","volume":"5","author":"Scott","year":"1962"},{"key":"S0022481200021320_ref024","doi-asserted-by":"publisher","DOI":"10.2307\/2026089"},{"key":"S0022481200021320_ref006","first-page":"557","volume":"41","author":"Friedman","year":"1976","journal-title":"Systems of second order arithmetic with restricted induction"},{"key":"S0022481200021320_ref020","volume-title":"Logic Colloquiu '80","author":"Simpson","year":"1982"},{"key":"S0022481200021320_ref018","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90030-2"},{"key":"S0022481200021320_ref016","first-page":"242","article-title":"On the degrees of complete extensions of arithmetic","volume":"7","author":"Scott","year":"1960","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200021320_ref003","first-page":"39","volume-title":"Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic","volume":"106","author":"Brown","year":"1990"},{"key":"S0022481200021320_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21755-9"},{"key":"S0022481200021320_ref010","first-page":"33","article-title":"classes and degrees of theories","volume":"173","author":"Jockush","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200021320_ref026","unstructured":"Yu X. , Measure theory in weak subsystems of second order arithmetic, Ph.D. Thesis , The Pennsylvania State University, University Park, PA, 1987."},{"key":"S0022481200021320_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90012-X"},{"key":"S0022481200021320_ref021","unstructured":"Simpson S. 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K. , Functional analysis in weak subsystems of second order arithmetic, Ph.D. Thesis , The Pennsylvania State University, University Park, PA, 1987."},{"key":"S0022481200021320_ref001","volume-title":"Logic and combinatorics","volume":"65","author":"Blass","year":"1986"},{"key":"S0022481200021320_ref019","first-page":"349","volume":"53","author":"Simpson","year":"1988","journal-title":"Partial realizations of Hilbert's program"},{"key":"S0022481200021320_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067653"},{"key":"S0022481200021320_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90014-S"},{"key":"S0022481200021320_ref025","first-page":"84","volume-title":"Das Kontinuum: Kritische Untersuchungen \u00fcber die Grundlagen der Analysis","author":"Weyl","year":"1917"},{"key":"S0022481200021320_ref023","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90074-6"},{"key":"S0022481200021320_ref005","first-page":"235","volume-title":"Proceedings of the International Congress of Mathematicians (Vancouver, 1974)","volume":"1","author":"Friedman","year":"1975"},{"key":"S0022481200021320_ref012","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"},{"key":"S0022481200021320_ref017","doi-asserted-by":"publisher","DOI":"10.2307\/1970028"},{"key":"S0022481200021320_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90066-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200021320","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T18:55:52Z","timestamp":1557946552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200021320\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,6]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,6]]}},"alternative-id":["S0022481200021320"],"URL":"https:\/\/doi.org\/10.2307\/2275219","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,6]]}}}