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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?

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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.

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  • $\begingroup$ Unless $\alpha$ is inaccessible (or wordly, etc.) $\endgroup$ – Pedro Sánchez Terraf Feb 19 '16 at 12:36

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