ordinal isomorphism theorem

Question

The only proof of the theorem stating that every well-order is order-isomorphic to a unique ordinal of which I am aware relies on the Axiom of Replacement. Is this necessary?

Answer 1

Yes, Replacement is necessary for this. For instance, it is straightforward to verify that $V_{\omega+\omega}$ is a model of all of ZFC except for Replacement (the same is true of $V_\alpha$ for any limit ordinal $\alpha>\omega$). In $V_{\omega+\omega}$, there are well-ordered sets of length $\omega+\omega$ and much longer (since for any set $X\in V_{\omega+\omega}$, every well-ordering of $X$ is also in $V_{\omega+\omega}$), but the ordinal $\omega+\omega$ does not exist.