  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.
