{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:33:55Z","timestamp":1759638835692},"reference-count":38,"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":6401,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Transitive extensional <jats:italic>well founded<\/jats:italic> relations provide an intuitionistic notion of <jats:italic>ordinals<\/jats:italic> which admits <jats:italic>transfinite induction<\/jats:italic>. However these ordinals are not <jats:italic>directed<\/jats:italic> and their successor operation is poorly behaved, leading to problems of functoriality.<\/jats:p><jats:p>We show how to make the successor monotone by introducing <jats:italic>plumpness<\/jats:italic>, which strengthens transitivity. This clarifies the traditional development of successors and unions, making it <jats:italic>intuitionistic<\/jats:italic>; even the (classical) proof of <jats:italic>trichotomy<\/jats:italic> is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very rapidly.<\/jats:p><jats:p>Directedness must be defined <jats:italic>hereditarily<\/jats:italic>. It is orthogonal to the other four conditions, and the <jats:italic>lower powerdomain<\/jats:italic> construction is shown to be the universal way of imposing it.<\/jats:p><jats:p>We treat ordinals as order-types, and develop a corresponding set theory similar to Osius' <jats:italic>transitive set objects<\/jats:italic>. This presents <jats:italic>Mostowski's theorem<\/jats:italic> as a reflection of categories, and set-theoretic union is a corollary of the <jats:italic>adjoint functor theorem<\/jats:italic>. Mostowski's theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank.<\/jats:p><jats:p>The comparison between sets and toposes is developed as far as the identification of <jats:italic>replacement<\/jats:italic> with completeness, and there are some suggestions for further work in this area.<\/jats:p><jats:p>Each notion of set or ordinal defines a <jats:italic>free algebra<\/jats:italic> for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation <jats:italic>s<\/jats:italic> satisfying conditions such as <jats:italic>x \u2264 sx<\/jats:italic>, monotonicity or <jats:italic>s<\/jats:italic>(<jats:italic>x \u2228 y<\/jats:italic>) \u2264 <jats:italic>sx \u2228 sy<\/jats:italic>.<\/jats:p><jats:p>Finally we discuss the <jats:italic>fixed point theorem<\/jats:italic> for a monotone endofunction <jats:italic>s<\/jats:italic> of a poset with least element and directed joins. This may be proved under each of a variety of <jats:italic>additional<\/jats:italic> hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result.<\/jats:p>","DOI":"10.2307\/2275781","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:59:03Z","timestamp":1146956343000},"page":"705-744","source":"Crossref","is-referenced-by-count":17,"title":["Intuitionistic sets and ordinals"],"prefix":"10.1017","volume":"61","author":[{"given":"Paul","family":"Taylor","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017084_ref031","first-page":"152","volume-title":"Logic in computer science 6","author":"Taylor","year":"1991"},{"key":"S0022481200017084_ref032","volume-title":"Practical foundations","author":"Taylor"},{"key":"S0022481200017084_ref005","volume-title":"On numbers and games","author":"Conway","year":"1976"},{"key":"S0022481200017084_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/092\/1003199"},{"key":"S0022481200017084_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF01458215"},{"key":"S0022481200017084_ref021","first-page":"442","article-title":"Well founded relations; generalisations of principles of induction and recursion","volume":"61","author":"Montague","year":"1955","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S0022481200017084_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF03015911"},{"key":"S0022481200017084_ref028","first-page":"217","volume-title":"Matematikerkongressen i Helsingfors den 4\u20137 Juli 1922, Den Funfe Skandinaviska Matematikenkongressen, Rodog\u00f6relse","author":"Skolem","year":"1922"},{"key":"S0022481200017084_ref022","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9478-5"},{"key":"S0022481200017084_ref001","volume-title":"Non-well-founded sets","author":"Aczel","year":"1988"},{"key":"S0022481200017084_ref014","first-page":"193","volume-title":"Category theory \u2014 Proceedings, Como, 1990","volume":"1488","author":"Johnstone","year":"1991"},{"key":"S0022481200017084_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457986"},{"key":"S0022481200017084_ref034","volume-title":"From Frege to G\u00f6del: a source book in mathematical logic, 1879\u20131931","author":"van Heijenoort","year":"1967"},{"key":"S0022481200017084_ref015","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511752483"},{"key":"S0022481200017084_ref013","volume-title":"Topos theory","author":"Johnstone","year":"1977"},{"key":"S0022481200017084_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(85)90103-3"},{"key":"S0022481200017084_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061825"},{"key":"S0022481200017084_ref019","doi-asserted-by":"publisher","DOI":"10.1007\/BF03025863"},{"key":"S0022481200017084_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(81)90016-4"},{"key":"S0022481200017084_ref009","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.24.12.556"},{"key":"S0022481200017084_ref018","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.52.6.1506"},{"key":"S0022481200017084_ref023","doi-asserted-by":"publisher","DOI":"10.4064\/fm-36-1-143-164"},{"key":"S0022481200017084_ref024","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(74)90032-2"},{"key":"S0022481200017084_ref026","doi-asserted-by":"publisher","DOI":"10.1007\/BF00370830"},{"key":"S0022481200017084_ref037","doi-asserted-by":"publisher","DOI":"10.1007\/BF01450054"},{"key":"S0022481200017084_ref017","volume-title":"Introduction to higher order categorical logic","volume":"7","author":"Lambek","year":"1986"},{"key":"S0022481200017084_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01446819"},{"key":"S0022481200017084_ref016","volume-title":"Surreal numbers","author":"Knuth","year":"1974"},{"key":"S0022481200017084_ref030","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0018351"},{"key":"S0022481200017084_ref033","volume-title":"Towards a unified theory of induction","author":"Taylor","year":"1996"},{"key":"S0022481200017084_ref020","first-page":"209","article-title":"Les antinomies de Russell et de Burali-Forti et le probl\u00e9me fondamental de la th\u00e9orie des ensembles, and Remarques sur la th\u00e9orie des ensembles et les antinomies cantoriennes. 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