I was trying to understand the union of a chain of cardinalities and I found this equation $$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question.
Can every set be expressed as the union of a chain of sets of lesser cardinality?
I dont understand how this could be possible if we assume there is no cardinality between $\mathbb N$ and $\mathbb R$.
In general, for any given cardinal $\kappa$, is there an expression for $$\bigcup_{\alpha<\kappa} \alpha$$
In the answers to the question, $\alpha$ is an ordinal smaller than $\kappa$, not a cardinal. Regardless to the continuum hypothesis you are correct that the only cardinals smaller than $\omega_1$ are finite and $\omega$.
But as an ordinal, $\omega_1$ is the increasing union of countable ordinals.