I'm studying Wadge reducibility and the associated hierarchy (restricted to Borel sets) both from Kechris and more extensive papers. Now, in an exercise Kechris says:
$[\emptyset]_W = \{\emptyset\}$, $[\mathcal{\omega^\omega}]_W = \{\mathcal{\omega^\omega}\}$ are at the bottom of the Wadge hierarchy. [...] It can be shown that level $\omega_1$ is occupied by sets that are $F_\sigma$ but not $G_\delta$.
I'm intereseted in the last statement. Why is it so? Where can I find the proof? The way it is put by Kechris makes me think that it is not an easy result.
More in general, given an ordinal $\alpha<\omega_1$, do we know at which level $\beta$ in the Wadge hierarchy (restricted to Borel sets) the sets in $\Sigma_\alpha^0(\mathcal{\omega^\omega})\setminus\Pi_\alpha^0(\omega^\omega)$ first appear? If not, can we at least pick a level $\beta$ which contains such sets?
The restriction of the Wadge hierarchy to Borel sets is not crucial to me, in case we can say something in the wider Wadge hierarchy (assuming AD for example).
Thanks!