From Wikipedia:
OD is not necessarily transitive, and need not be a model of ZFC.
This obviously means that, assuming ZFC is consistent, there is a model $M \models \mathrm{ZFC}$ so that $\mathrm{OD}^M \not \models \mathrm{ZFC}$, and yet $M \models M \models \mathrm{ZFC}$ by absoluteness of $\Delta_0$-formulae, and so $M \models \mathrm{ZFC} + V \neq \mathrm{OD}$.
Now, if $V \neq \mathrm{OD}$, then $V_\alpha \cap \mathrm{OD} \subset V_\alpha$ for some $\alpha$ - if such an $\alpha$, what is it or bounds on it? Can we find models $M$ in which the least such $\alpha$ is unboundedly large (i.e. V can be arbitrarily close to OD while not being equal), or would instead the least $\alpha$ be recursive, countable, etc? If I remember correctly, ordinal definability is not absolute, so what would the least $\alpha$ such that $\mathrm{OD}^{V_\alpha} \neq V_\alpha \cap \mathrm{OD}$?
