Recently, i have studied the proof of why the set of all countable ordinals $\Gamma$ is an ordinal and why it is an uncountable. (Basically, since it contains all countable ordinals as predecessors, it cannot contain itself, by definition).
Authors immediately concluded that since this ordinal is uncountable, its cardinal is $\aleph_{1}$.
However, i want to clearly see that, indeed $|\omega_{1}| = \aleph_{1}$, by creating some isomorphism between $\omega_{1}$ and the set of real numbers.
And I don't even understand how to begin to proof that or is it even possible? Does continuum hypothesis has something to do with my proposition? Or simply, does my answer has answer in the term of ZFC axioms?
If my question is quite deep and goes beyond standard axioms of set theory, let me know. I'm not advanced, i was just intrigued by that question.