Exponentiating limit ordinals

Question

I have been reading about ordinal arithmetic and I came across this definition for ordinal exponentiation, when the exponent is a limit ordinal and the base isn't. Here is the definition: $$\begin{array}{lcl} \alpha^0 &= & 0' \\ \alpha^{\beta'} &= &(\alpha^\beta) \times \alpha \\ \alpha^{\lambda} &= & {\bigcup \{\alpha^\beta : \beta < \lambda\} \text{ ($\lambda$ a limit ordinal)}} \end{array}$$ The definition I am talking about is the last one. I was wondering if there is a similar definition for when the base is also a limit ordinal. Any help on this would be greatly appreciated.

Answer 1

The definition you've written is usually given in that exact form for all ordinals $\alpha$, nor there seems to be any material reason to make a special case for $\alpha=\gamma'$. For instance, that's the definition in wikipedia and in Set theory by Thomas Jech. But it should be the same in every book ever written.