Ordinal exponentiation (${\alpha }^{\beta }$)

Ordinal exponentiation is defined recursively byfor nonzero limit $\lambda$. Transfinite induction gives

The ordinal ${\epsilon }_0$ is the least fixed point of $\alpha \mapsto {\omega }^{\alpha }$. It is the supremum of $1,\omega ,{\omega }^{\omega },{\omega }^{{\omega }^{\omega }},\dots$, and is countable.