Examples of $\omega_{1}<\alpha<\omega_{2}$ definable

Question

Can you think of any examples of definable ordinals between $\omega_{1}<\alpha<\omega_{2}$?

I am trying to show that countable $M\prec (H(\aleph_{2},\in)$ contains ordinals $>\omega_{1}$.

Is $2^{\omega_{1}}$ one? Thanks

Answer 1

Even if you don't show any direct example, it is easy to show that.

Note that your structure doesn't have a maximal ordinal. So if $\omega_1$ is in $M$ it cannot be the maximal one. Then to show it is definable is not hard either. For example $\omega_1+\omega$ is definable.

Finally, as Arthur points out, $2^\delta=\sup\{2^\beta\mid\beta<\delta\}$ for a limit ordinal $\delta$, therefore $2^{\omega_1}=\sup\{2^\alpha\mid\alpha<\omega_1\}$. It can be shown by induction that $\alpha^\beta$ is countable, if both ordinals are countable. Therefore $2^{\omega_1}$ is the supremum of countable ordinals, but it has to be uncountable and therefore it must be $\omega_1$ itself.