On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos
Abstract
In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2009
- DOI:
- 10.48550/arXiv.0911.0068
- arXiv:
- arXiv:0911.0068
- Bibcode:
- 2009arXiv0911.0068B
- Keywords:
-
- Mathematics - Logic;
- Mathematics - Category Theory;
- 03G30;
- 03G10
- E-Print:
- doi:10.1016/j.tcs.2011.12.005