Let $N$ be a model of first-order Peano arithmetic, and let $\sigma$ be its order-type. Does it follow that there is a (non-transitive, expect when $M$ is the standard model) $ZFC$-model $M$ such that $\omega^{M}=\sigma$? What if $PA$ is strengthened to true arithmetic (i.e. the theory of $(\mathbb{N},+,\cdot)$ or weakened to a subtheory such as $I\Sigma_{1}$, $I\Delta_{0}+EXP$, $I\Delta_{0}$ or $IOpen$? Or when we replace $ZFC$ with something weaker like $KP$?