{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T14:02:51Z","timestamp":1748959371738},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5215,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,12]]},"abstract":"<jats:p>Call a family <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012731_inline1\" \/> of subsets of a set <jats:italic>E<\/jats:italic> inductive if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012731_inline2\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012731_inline1\" \/> is <jats:italic>closed under unions with disjoint singletons<\/jats:italic>, that is, if<\/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=\"S0022481200012731_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>A <jats:italic>Frege structure<\/jats:italic> is a pair (<jats:italic>E<\/jats:italic>, \u03bd) with \u03bd a map to <jats:italic>E<\/jats:italic> whose domain dom(\u03bd) is an inductive family of subsets of <jats:italic>E<\/jats:italic> such that<\/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=\"S0022481200012731_Uequ2\" \/><\/jats:disp-formula><\/jats:p><jats:p>In [2] it is shown in a constructive setting that each Frege structure determines a subset which is the domain of a model of Peano's axioms. In this note we establish, within the same constructive setting, three facts. First, we show that the least inductive family of subsets of a set <jats:italic>E<\/jats:italic> is precisely the family of <jats:italic>decidable Kuratowski finite<\/jats:italic> subsets of <jats:italic>E<\/jats:italic>. Secondly, we establish that the procedure presented in [2] can be reversed, that is, any set containing the domain of a model of Peano's axioms determines a map which turns the set into a <jats:italic>minimal<\/jats:italic> Frege structure: here by a minimal Frege structure is meant one in which dom(\u03bd) is the least inductive family of subsets of <jats:italic>E<\/jats:italic>. And finally, we show that the procedures leading from minimal Frege structures to models of Peano's axioms and vice-versa are <jats:italic>mutually inverse<\/jats:italic>. It follows that the postulation of a (minimal) Frege structure is constructively equivalent to the postulation of a model of Peano's axioms.<\/jats:p><jats:p>All arguments will be formulated within constructive (intuitionistic) set theory.<\/jats:p>","DOI":"10.2307\/2586795","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:03:57Z","timestamp":1146938637000},"page":"1552-1556","source":"Crossref","is-referenced-by-count":2,"title":["Finite sets and frege structures"],"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":"S0022481200012731_ref001","volume-title":"Toposes and local set theories: An introduction","author":"Bell","year":"1988"},{"key":"S0022481200012731_ref002","first-page":"486","volume":"64","author":"Bell","year":"1999","journal-title":"Frege's theorem in a constructive setting"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200012731","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T19:57:48Z","timestamp":1557431868000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200012731\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,12]]},"references-count":2,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1999,12]]}},"alternative-id":["S0022481200012731"],"URL":"https:\/\/doi.org\/10.2307\/2586795","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,12]]}}}