For a given $\aleph_\alpha$, what is the minimal ordinal possible such that $|\beta| = \aleph_\alpha$? More precisely, assume we have a model $N$ of ZFC. $N$ could be an extension of many different models. We have the class of ordinals, which is absolute. Some ordinals in $N$ could be equal to $\omega_1^M$ for some $N = M[G]$, but definitely not $(\omega + 2)^N$ for example. Obviously, there is a minimal ordinal in $N$, which could have been $\omega_1$ in a base model. How do we know which one it is? And for $\omega_\alpha$ in general?
In some way it feels like this ordinal is very "characteristic" of this cardinal, so I wonder if this concept has a name. And what can we say in general about this ordinal?