On the failure of fixed-point theorems for chain-complete lattices in the effective topos

We show that in the effective topos there is a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. An immediate consequence of this is that both Tarski’s theorem for chain-complete posets and the Bourbaki–Witt theorem have no constructive (topos-valid) proofs.