TITLE: How to define countability of $\omega^{\omega}$ and $\omega_1$? in set theory?
QUESTION [7 upvotes]: How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two numbers? 
Edit: changing the question into how $\omega^{\omega}$ can be shown countable, and how other countable ordinals can be shown to be countable.

REPLY [2 votes]: Generally speaking, the best course of action when trying to show that $\alpha$ is a countable ordinal, is to show that:

$\alpha = \delta+n$ where $\delta$ is a limit ordinal (if $n=0$ then $\alpha=\delta$);
$\delta$ is the limit of $\{f(\beta)\mid \beta<\gamma\}$ for some countable $\gamma$; and
$f(\beta)$ is countable is countable whenever $\beta$ is countable.

For example, $\epsilon_0$ is the limit of $\omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots$. We can show that this is the limit of the recursive function $f(n+1)=\omega^{f(n)}$ and $f(0)=\omega$. We can further show that if $\beta$ is countable then $\omega^\beta$ is also countable (note that this is ordinal exponentiation) and therefore $\epsilon_0$ is the countable limit of countable ordinals and thus countable.
The same method can be applied to general ordinals, although obtaining such $f$ is often harder than it is in the case of $\epsilon_0$.
One caveat is that this is true in ZFC but not necessarily in ZF, where a countable union of countable ordinals might not be countable.