Biggest countable ordinal number

Question

I need to find biggest countable ordinal number. But I am sure there is no one, if I understand proofs and definitions correctly. So here are my idea: Suppose there is biggest countable ordinal number $\alpha$. That means that next ordinal number is smallest uncountable ordinal number, $\omega_1$, and $\omega_1 = \alpha+1$. But $\omega_1$ is limit ordinal number, because if not, there would be something like: $\omega_1 = \alpha+\omega^\omega+\omega+...$ and $\alpha$ would be smallest uncountable ordinal. Are are thought correct, or there exists biggest countable ordinal number? If it here, how do I find it?

Answer 1

There isn't one. If $\alpha$ is a countable ordinal number, then $\alpha+1$ is also a countable ordinal number which is strictly bigger, so there cannot be a largest one.

Even if we have $\alpha_0 < \alpha_1 < \alpha_2 \ldots$, all countable ordinal numbers, then $\cup_n \alpha_n$ is still a countable ordinal number bigger than all of them (as a countable union of countable sets is countable).

This shows that the first uncountable ordinal $\omega_1$ must be a limit as well, and it has so-called co-finality $\omega_1$: it is not a limit of only countably many smaller ordinals.