Answer by Zetapology for Do these ordinals exist?

You can guarantee $F_n(\alpha)$ is countable.

Assume the contrary. There is a first-order formula for every countable ordinal $\phi$ such that $(V\models\phi(S,F_1(\alpha)...))\Leftrightarrow S=\alpha$

Clearly there is a bijection between the set of all these formulas and $\omega_1$. But using Godel numbers, this set is countable. Therefore we have a Contradiction.