In this question I am supposing that both $\alpha$ and $\beta$ are ordinals.
My definition of an ordinal is that: $x$ is an ordinal if $x$ is well-ordered by $\in$ and $x$ is $\in$-transitive.
So far from definitions I have that $\alpha^\beta$ = sup{$\alpha ^ \gamma$ | $\gamma \in \beta$} (where $\in$ = <) and $\beta$ is a limit ordinal.
I also know that since $\omega_1$ is the first uncountable ordinal, then $\alpha$ $\in$ $\omega_1$ is an ordinal. (Not sure if this is correct)
Then I think I just need to prove that $\alpha ^ \beta$ $\in$ $\omega_1$? But I'm not sure how to do that.
Thank you in advance
Best Answer
HINT:
Prove by induction that the successor of a countable ordinal is countable; conclude that addition of countable ordinals is countable; conclude that a product of countable ordinals is countable; and finally conclude that the exponentiation of two countable ordinals is countable.
The idea behind all of these is the same: countable union of countable sets is countable.
(One can also do that directly, without appealing to the axiom of choice as above, but it does make life simpler.)