Can we define ordinals such that the following sentences are independent of ZFC?

Question

Can we explicitly define two ordinals $\alpha$ and $\beta$ in the language of $\{\in\}$ such that the following hold?

1. ZFC proves that $\alpha$ and $\beta$ exist.
2. ZFC proves that $\beth_\beta \neq \aleph_\alpha$
3. The following statements are independent of ZFC: $$\aleph_\alpha \leq \beth_\beta,\qquad\beth_\beta \leq \aleph_\alpha$$

Answer 1

Doesn't this happen if $\alpha=\omega$ and $\beta=1$?

Fact 2 follows from Konig's theorem and fact 3 can be proved by forcing arguments.