On the BourbakiWitt principle in toposes
Abstract
The BourbakiWitt principle states that any progressive map on a chaincomplete 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 BourbakiWitt 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.
 Publication:

Mathematical Proceedings of the Cambridge Philosophical Society
 Pub Date:
 July 2013
 DOI:
 10.1017/S0305004113000108
 arXiv:
 arXiv:1201.0340
 Bibcode:
 2013MPCPS.155...87B
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Logic;
 06B23;
 03G30;
 03F55
 EPrint:
 Math. Proc. Cam. Phil. Soc. 155 (2013), no. 1, 8799