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On the Bourbaki–Witt principle in toposes

  • ANDREJ BAUER (a1) and PETER LEFANU LUMSDAINE (a2)

Abstract

The Bourbaki–Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically.

We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki–Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.

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[1]Aczel, P. and Rathjen, M. Notes on constructive set theory. Technical report, Institut Mittag–Leffler Preprint (2001).
[2]Bauer, A.On the failure of fixed-point theorems for chain-complete lattices in the effective topos. Theoret. Comput. Sci. 430 (2012), 4350. Available from http://arxiv.org/abs/0911.0068arXiv:0911.0068.
[3]Bourbaki, N.Sur le théorème de Zorn. Arch. Math. 2 (6) (November 1949), 434437.
[4]Dacar, F.Suprema of families of closure operators. Seminar for foundations of mathematics and theoretical computer science (November 2008). Faculty of Mathematics and Physics, University of Ljubljana, Slovenia.
[5]Dacar, F. The join-induction principle for closure operators on dcpos. Available from http://dis.ijs.si/France/ (January 2009).
[6]Lambek, J. and Scott, P. J.Introduction to higher order categorical logic. Cambridge Studies in Advanced Math., vol. 7 (Cambridge University Press, 1986).
[7]Lang, S.Algebra. Graduate Texts in Mathematics, vol. 211 (Springer-Verlag, New York, third edition, 2002).
[8]Mac Lane, S. and Moerdijk, I.Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer-Verlag, 1992).
[9]Pataraia, D. A constructive proof of Tarski's fixed-point theorem for dcpos. 65th Peripatetic Seminar on Sheaves and Logic (November 1997).
[10]Rogers, H.Theory of Recursive Functions and Effective Computability (MIT Press, third edition, 1992).
[11]Rosolini, G.Un modello per la teoria intuizionista degli insiemi. In Bernardi, C. and Pagli, P., editors, Atti degli Incontri di Logica Matematica, pp. 227230, Siena, 1982. English translation available at http://www.disi.unige.it/person/RosoliniG/RosoliniG_modtii_eng.pdf.
[12]Tarski, A.A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5 (2) (1955), 285309.
[13]Taylor, P.Intuitionistic sets and ordinals. J. Symbolic Logic 61 (3) (1996), 705744.
[14]van Oosten, J.Realizability: An Introduction to its Categorical Side. Studies in Logic and the Foundations of Mathematics. vol. 152 (Elsevier, 2008).
[15]Witt, E.Beweisstudien zum Satz von M. Zorn. Math. Nachr. 4 (1951), 434438.

On the Bourbaki–Witt principle in toposes

  • ANDREJ BAUER (a1) and PETER LEFANU LUMSDAINE (a2)

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