TITLE: ordinal isomorphism theorem
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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.