{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T14:02:49Z","timestamp":1748959369311},"reference-count":1,"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":5398,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,6]]},"abstract":"<jats:p>By <jats:italic>Frege's Theorem<\/jats:italic> is meant the result, implicit in Frege's <jats:italic>Grundlagen<\/jats:italic>, that, for any set <jats:italic>E<\/jats:italic>, if there exists a map \u03c5 from the power set of <jats:italic>E<\/jats:italic> to <jats:italic>E<\/jats:italic> satisfying the condition<\/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=\"S0022481200013438_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>then <jats:italic>E<\/jats:italic> has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map \u03c5 be defined on the family of (Kuratowski) <jats:italic>finite<\/jats:italic> subsets of the set <jats:italic>E<\/jats:italic>, and secondly, the argument will be <jats:italic>constructive<\/jats:italic>, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the following<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc>. <jats:italic>Let \u03c5 be a map with domain a family of subsets of a set E to E satisfying the following conditions<\/jats:italic>:<\/jats:p><jats:p>(i) \u00f8 \u03f5<jats:italic>dom<\/jats:italic>(\u03c5)<\/jats:p><jats:p>(ii)\u2200<jats:italic>U<\/jats:italic> \u03f5<jats:italic>dom<\/jats:italic>(\u03c5)\u2200<jats:italic>x<\/jats:italic> \u03f5 <jats:italic>E<\/jats:italic> \u2212 <jats:italic>U<\/jats:italic><jats:italic>U<\/jats:italic> \u222a <jats:italic>x<\/jats:italic> \u03f5<jats:italic>dom<\/jats:italic>(\u03c5)<\/jats:p><jats:p>(iii)\u2200<jats:italic>UV<\/jats:italic> \u03f5<jats:italic>dom<\/jats:italic>(5) \u03c5(<jats:italic>U<\/jats:italic>) = \u03c5(<jats:italic>V<\/jats:italic>) \u21d4 <jats:italic>U<\/jats:italic> \u2248 <jats:italic>V<\/jats:italic>.<\/jats:p><jats:p><jats:italic>Then we can define a subset N of E which is the domain of a model of Peano's axioms<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2586481","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:03:01Z","timestamp":1146938581000},"page":"486-488","source":"Crossref","is-referenced-by-count":5,"title":["Frege's theorem in a constructive setting"],"prefix":"10.1017","volume":"64","author":[{"given":"John L.","family":"Bell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013438_ref001","first-page":"209\u2013220","volume":"60","author":"Bell","year":"1995","journal-title":"Type reducing correspondences and well-orderings: Frege's and Zermelo's constructions re-examined"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013438","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T21:18:31Z","timestamp":1557436711000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013438\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,6]]},"references-count":1,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1999,6]]}},"alternative-id":["S0022481200013438"],"URL":"https:\/\/doi.org\/10.2307\/2586481","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,6]]}}}