What is a useful definition of an ordinal in Grothendieck toposes. Useful in this context means that I would like to construct fixed points of monotone maps on complete lattices using iteration from $\bot$. Can this be done? I know one can prove that there exists the least fixed point by constructing it as the intersection of all the pre-fixed points.
To define the chains I presumably need some form of well-founded induction principle, but the definitions of ordinals that are classically equivalent turn out to not be equivalent intuitionistically, and I can't find a proper reference explaining the relationship among them.
Any references explaining these issues are also very welcome.