Okay my understanding so far is that: We have ordinal exponentiation which I think is a normal function.. Now the sequence $ \lbrace n | n < \omega \rbrace$ is increasing. Let $f: \omega \to \omega, \alpha ^ \omega (\alpha \text{ is finite})$. Then $\lim_{\beta < \omega} \alpha ^ \omega = \lim_{\beta < \omega} n = \omega$.
There's also this I've seen where $\alpha ^ \omega = \bigcup_{\beta < \omega} \alpha ^ \beta = \omega$.
I think that $\alpha ^ \omega = \bigcup_{\beta < \omega} \alpha ^ \beta $ comes from $\alpha^\omega$ with $\alpha < \omega$ being an increasing sequence with a unique limit. But why does this equal $\omega$