some classmates and I are attacking this problem, which is exercise 3.10 of Jech's "Set theory" book:
$\omega_{\alpha+1}$ is a projection of $\wp(\omega_{\alpha})$
So, we need to find an onto function from $\wp({\omega_{\alpha}})$ to $\omega_{\alpha+1}$. We know that $|\omega_{\alpha}\times\omega_{\alpha}|=\omega_{\alpha}$, therefore we search for a surjective function from $\wp(\omega_{\alpha}\times\omega_{\alpha})$ to $\omega_{\alpha+1}$.
The book's hint goes as follows: "if $R\subseteq\omega_{\alpha}\times\omega_{\alpha}$ is a well-order, let $f(R)$ be its order type". However we can't have a good idea about how to proceed. Can anyone please help us?
Ask yourselves, what are the elements of $\omega_{\alpha+1}$? These are ordinals whose cardinality is at most $\aleph_\alpha$. Therefore each such ordinal has an injection into $\omega_\alpha$.
So for each $\eta<\omega_{\alpha+1}$, there is some $R\subseteq\omega_\alpha\times\omega_\alpha$ which is a well-ordering of its field (domain and range) with order type $\eta$.