I've got a most probably silly question, but I can't find the answer to it:
If $\alpha$ is a countable limit ordinal, how can we be sure that there exist ordinals $\alpha_n$ such that $\alpha = \bigcup \{\alpha_n\mid n\in\omega\}$ (rather than the usual $\alpha = \bigcup \{\beta\mid \beta<\alpha\}$)?
Cheers!
Well, first of all $\{\beta\mid\beta<\alpha\}=\alpha$ is countable, therefore it can be enumerated as $\alpha_n$'s. But I suppose that you want to ask about a strictly increasing sequence.
First enumerate $\alpha$ as $\{\beta_n\mid n<\omega\}$. Then by induction define $\alpha_n$, where $\alpha_{n+1}=\beta_k$ where $k=\min\{m\mid\alpha_n<\beta_m\}$. You can now show that this is a strictly increasing and cofinal sequence.