How many limits of limits are there in $\omega_1$?

Question

We know there are $\omega_1$-many limit ordinals less than $\omega_1$. What about ordinals that are limits of limit ordinals? Are there also $\omega_1$-many of these?

Answer 1

Yes! For every ordinal $\alpha$, there is a limit of limits less than $\omega_1$ but greater than $\alpha$. So by regularity, there cannot be only countably many.

This argument works for any unbounded set, actually. Proving that sets are unbounded can be done manually, or with the help of some theory of stationary and club sets.

Answer 2

Yes, just note that for every $\alpha$, $\alpha+\omega^2$ is a limit of limit ordinals. And while the function $\alpha\mapsto\alpha+\omega^2$ is not injective, each fiber is countable. So there has to be uncountably many values the function takes below $\omega_1$.