How does one prove that there are uncountable number of countable ordinals?

Question

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Uncountability of countable ordinals

How does one prove that there are uncountable number of countable ordinals? Obviously, there are equal to or more than countable number of ordinals, but not sure how to prove there are uncountable number of countable numbers. Maybe, should I just assume that as uncountable ordinal has uncountable elements as a set, there should be uncountable number of countable ordinals?

Answer 1

If $A$ is a countable set of countable ordinals, let $\beta=\sup\{\alpha+1:\alpha\in A\}$; then $\beta$ is countable, and $\alpha<\beta$ for every $\alpha\in A$. Thus, the set of countable ordinals cannot be countable.

Added: To see that $\beta$ is countable, just note that $\alpha+1$ is clearly countable if $\alpha$ is, so $\beta=\bigcup_{\alpha\in A}(\alpha+1)$ is a countable union of countable sets. This actually also covers the existence of $\beta$: given the set $\{\alpha+1:\alpha\in A\}$, whose existence follows from the axiom schema of replacement and the fact that $A$ is a set, the union axiom ensures the existence of $\sup\beta$.

Answer 2

Define $\Omega$ to be the smallest uncountable ordinal. It exists because uncountable ordinals exist, and ordinals are well-ordered (any nonempty class of ordinals has a smallest member).

Now $\Omega$ is the set of all countable ordinals, but it is itself uncountable by construction.