What is the cardinality of $\Omega$?

Consider,

$\Omega=$Set of all countable ordinals.

Then, $\Omega$ is uncountable.

Proof: Suppose, $\Omega$ is countable, then it has a enumeration say $(\alpha_n) _{n\in{\mathbb{N}}}$.

Then, $ \alpha = \sum_{n\in {\mathbb{N}}}{\alpha_n} +1$ is a countable ordinal but $ \alpha\neq{\alpha_n}$ for any $n\in {\mathbb{N}}$. Hence, the contradiction.

This implies $\aleph_0 < card(\Omega) $

My Question: What is the cardinality of $\Omega$ ?

Please explain in some details.

If the question is already answered, please provide the link. Thanks

It is the first, i.e., the smallest uncountable ordinal, since every ordinal smaller than this ordinal is countable.

If you assume Continuum Hypothesis, then the cardinality is same as the cardinality of real numbers. Without $\mathbf{CH}$, you can not say much about it$\dots$ It is known that without $\mathbf{CH}$ there can be cardinals as much as you want between $|\mathbb N|$ and $|\mathbb{R}|=|\mathcal{P}(\mathbb{N})|$.