Countable fixed point of a normal ordinal function ($\alpha ={\sup }_{n<\omega }{\Phi }^n(0)$)

If a normal function on an ordinal begins at a countable value and preserves countability from $\Phi (\xi )$ to $\Phi (\xi +1)$, transfinite induction shows it maps all countable ordinals to countable ordinals. At countable limits this uses countable unions of countable sets, hence axiom of countable choice in the usual proof. Iteration from zero either stabilizes at a countable fixed point or has a countable limit fixed by continuity. Applying this to ${\omega }^{\alpha }$ gives the countable epsilon number ${ϵ}_0$.