# Do the well-ordered sets of a model of ZF form a model of ZFC?

Let $$M$$ be a model of ZF. Consider the substructure $$N$$ of $$M$$ consisting of those elements that can be well ordered. Is $$N$$ a model of ZFC?

$$N$$ may not even be transitive, since a singleton can always be well-ordered. It's more meaningful to consider the class $$HWO$$ of hereditarily well-orderable sets. Even then, Exercise II.2.16 in Kunen says $$HWO$$ may not satisfy replacement or power set.
Actually, it seems that $$HWO$$ never satisfies power set axiom unless $$\mathsf{AC}$$ already holds in $$V$$. As remarked in Kunen after the above exercise, $$\mathcal{P}(\kappa)\subseteq HWO$$ for any $$\kappa$$, so if $$HWO$$ satisfies power set axiom we must have $$\mathcal{P}(\kappa)\in HWO$$, so $$\mathcal{P}(\kappa)$$ is well-orderable for any $$\kappa$$. By Kunen Exercise I.12.17, this implies $$\mathsf{AC}$$. It is still interesting to see when does $$HWO$$ satisfy replacement.
• @Carla_ Most of the natural models studied in set theory are transitive. A random non-transitive model $(M,\in)$ may not even satisfy extensionality. Indeed, as long as there exists some non-well-orderable set $x$, we have $\{x\}\in WO$, so extensionality fails in $WO$ because $\{x\}$ and $\emptyset$ have the same elements but are different sets. Commented Aug 3 at 14:33