Ordinal exponentiation
= Ordinal exponentiation
{title2=$\alpha^\beta$}
{wiki}
Ordinal exponentiation is defined recursively by
$$
\alpha^0=1,\qquad
\alpha^{\beta+1}=\alpha^\beta\alpha,\qquad
\alpha^\lambda=\sup_{\xi<\lambda}\alpha^\xi
$$
for nonzero limit $\lambda$. Transfinite induction gives
$$
\alpha^{\beta+\gamma}=\alpha^\beta\alpha^\gamma.
$$