{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T05:12:10Z","timestamp":1697951530230},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14072,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,9]]},"abstract":"<jats:p>The aim of this paper is to correct an error in the proof in [2] of a strengthened version of Hilbert's second \u03b5-theorem.<\/jats:p><jats:p>Hilbert's original theorem says (effectively) that formulae of the form \u2203<jats:italic>xA<\/jats:italic> \u2192 <jats:italic>A<\/jats:italic>(\u03b5<jats:italic>xA<\/jats:italic>) can be eliminated from deductions of \u2208-free sentences <jats:italic>B<\/jats:italic> (i.e. those which do not contain the \u03b5-symbol) from collections <jats:italic>X<\/jats:italic> of \u03b5-free sentences. The extended version (Theorem III. 11 of [2]) says that, in addition, instances of Ackermann's schema <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052993_inline1\" \/> can also be eliminated.<\/jats:p><jats:p>The error in [2] occurs in the proof of Theorem III.7 and is described as follows. If <jats:italic>A<\/jats:italic>\u2032 is the formula obtained from <jats:italic>A<\/jats:italic> by replacing every occurrence of \u2203<jats:italic>yB<\/jats:italic> by <jats:italic>B<\/jats:italic>(\u03b5<jats:italic>yB<\/jats:italic>) (or \u2200<jats:italic>yC<\/jats:italic> by <jats:italic>C<\/jats:italic>(\u03b5<jats:italic>yB<\/jats:italic>) if <jats:italic>B<\/jats:italic> is of the form \u00ac <jats:italic>C<\/jats:italic>), and if <jats:italic>A<\/jats:italic> is a <jats:italic>Q<\/jats:italic>3 or <jats:italic>Q<\/jats:italic>4-axiom then, contrary to the claim on p. 73 of [2], <jats:italic>A<\/jats:italic>\u2032 is not necessarily an axiom of the same form. For example, if <jats:italic>A<\/jats:italic> is the <jats:italic>Q<\/jats:italic>3-axiom<\/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=\"S0022481200052993_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where the rank of \u03b5<jats:italic>x<\/jats:italic>\u2203<jats:italic>yD<\/jats:italic>(<jats:italic>x, y<\/jats:italic>) is \u2264 the rank of <jats:italic>\u03b5yB<\/jats:italic> (see p. 70) and where <jats:italic>B<\/jats:italic> is <jats:italic>D<\/jats:italic>(<jats:italic>t,y<\/jats:italic>), then <jats:italic>A<\/jats:italic>\u2032 is<\/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=\"S0022481200052993_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>which is not a <jats:italic>Q<\/jats:italic>3-axiom. In other words, the notion of rank as defined in [2] is inadequate for the proof of Theorem III.11.<\/jats:p>","DOI":"10.2307\/2272162","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:36:35Z","timestamp":1146951395000},"page":"393-397","source":"Crossref","is-referenced-by-count":5,"title":["On an extension of Hilbert's second \u03b5-theorem"],"prefix":"10.1017","volume":"40","author":[{"given":"T. B.","family":"Flannagan","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200052993_ref004","volume-title":"The mathematics of metamathematics","author":"Rasiowa","year":"1968"},{"key":"S0022481200052993_ref003","unstructured":"Leisenring A. C. , Ph.D. Thesis, London, 1967."},{"key":"S0022481200052993_ref002","volume-title":"Mathematical logic and Hilbert's \u03b5-symbol","author":"Leisenring","year":"1969"},{"key":"S0022481200052993_ref001","volume-title":"Grundlagen der Mathematik","volume":"2","author":"Hilbert","year":"1939"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200052993","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T19:21:09Z","timestamp":1559157669000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200052993\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200052993"],"URL":"https:\/\/doi.org\/10.2307\/2272162","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}