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On the Bourbaki–Witt principle in toposes

Published online by Cambridge University Press:  18 March 2013

ANDREJ BAUER  and
PETER LEFANU LUMSDAINE
ANDREJ BAUER
Affiliation:
University of Ljubljana, Slovenia. e-mail: Andrej.Bauer@andrej.com
PETER LEFANU LUMSDAINE
Affiliation:
Dalhousie University, Halifax, Canada. e-mail: p.l.lumsdaine@gmail.com
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 87 - 99
Copyright
Copyright © Cambridge Philosophical Society 2013 

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