Do all topos have "enough ordinals" to get fixpoint by iterations – mathoverflow.net

Working internally in an elementary topos $\mathcal{E}$, let $X$ be an object of $\mathcal{E}$, $\mathcal{P}(X)$ its power object and consider a morphism $S: \mathcal{P}(X) \to \mathcal{P}(X)$ which ...