$\omega_1^{CK} - \omega$ - infinite or finite set? And boundary

Question

I am curious whether $\omega_1^{CK} - \omega$ would result in a finite set or infinite set. Does anyone know what happens?

Edit: OK, let me add one more question: Suppose that we take $\omega \cdot \kappa$ where $\kappa$ is some ordinal. What would be $\kappa$ that is the least ordinal so that $\omega_1^{CK} < \omega \cdot \kappa$? (.. and this is already answered.)

Answer 1

$\omega_1^{\text{CK}}$ cannot be a successor ordinal, and therefore $\omega + n \in \omega_1^{\text{CK}} \setminus \omega$ for all $n \in \omega$.

Answer 2

It is not only it is not a successor, but also $\omega_1^{CK}\setminus\omega$ is isomorphic to $\omega_1^{CK}$.

It is not hard to show that $\omega_1^{CK}$ is closed under products, so $\kappa=\omega_1^{CK}+1$.