Cardinal numbers bigger than $\omega$

Question

In my set theory textbook it is left for an exercise to prove that all natural numbers and $\omega$ are cardinal numbers and that every other bigger ordinal number up to and including $\epsilon_0:=sup ${1, $\omega$, $\omega^\omega$, ...} is not a cardinal number. I don't understand how can it be that we don't have any cardinals between $\omega$ and $\epsilon_0$, it seems like a huge gap. Also, there is a theorem that says there is a unique cardinal number $\alpha$ for every set a, such that $\alpha$ is equipotent with a. Does this mean that all ordinals from $\omega$ to $\epsilon_0$ are equipotent, i.e. belong to the same one equivalence class?

Answer 1

The first uncountable ordinal, is the first cardinal number after $\omega$, everything in between is countable and therefore is not a cardinal number.