I'm having trouble understanding initial ordinals. In particular, I can't prove a seemingly trivial theorem about them.
Def: An ordinal $\kappa$ is an initial ordinal iff $\forall \delta < \kappa \, \vert \delta \vert < \vert \kappa \vert$.
Proposition. For all ordinals $\alpha$ there is an initial ordinal $\kappa$ such that $\vert \alpha \vert = \vert \kappa \vert$. This proposition does not us AC.
I have tried so many different permutations for this proposition, but none seem to work. This means there is something that I don't understand concerning initial ordinals. My first approach was to let $\kappa$ be the least ordinal such that $\vert \alpha \vert = \vert \kappa \vert$, and then show that $\kappa$ is initial. The proof must be something along these lines. Any help would be greatly appreciated.