On the Failure of Fixed-point Theorems for Chain-complete Lattices in the Effective Topos
- 23 January 2009
- Constructive math, Publications
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.
Download: fixed-points.pdf
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Very nice paper!
I think there may be a typo in the first paragraph of Sect. 2, by the way: you define that X is discrete if the diagonal map X --> X^{\nabla 2} is constant. But if internally this is the condition that every map fron \nabla 2 to X is constant, i.e. in the image of the diagonal map, should the original condition be that the diagonal map is epimorphic?
Thank you for spotting that one. It should say that the diagonal map is an isomorphism, or equivalently epimorphism, as you suggest.