It appears to be quite a folklore result that under AD we know which small well-orderable cardinals (for my purposes I mean below $\omega_\omega$) are regular, namely only $\omega,\omega_1,\omega_2$. My question regards proofs of these results:
Where could I find the proof that under AD only the mentioned cardinals are regular?
I can recall that $\omega_1$ is regular thanks to countable choice for reals, and I guess regularity of $\omega_2$ might follow from measurability, but I'm mostly interested in a proof of singularity of all larger cardinals up to $\omega_\omega$.
Thanks in advance.
A proof can be found in Kanamori's book The higher infinite. The result follows from work of Solovay on the theory of uniform indiscernibles and work of Martin on projective scales. Look at sections 14, 28, and 30 of the book.
A different approach (using the infinite partition properties of $\omega_1$ and $\omega_2$, themselves due to Solovay and Martin) , is due to Kleinberg, and the details can be found on his book Infinitary combinatorics and the axiom of determinateness. For modern extensions of Kleinberg's results, see here.
A related open problem (probably due to Steve Jackson): Under determinacy, is it true that the cofinality function is non-decreasing when restricted to double successor cardinals below $\Theta$?
(The answer is obviously negative if we do not restrict our attention to successor cardinals. It is negative as well if we do not further restrict to double successors. For instance, $\aleph_{\omega 2 +1}$ is regular, while $\aleph_{\omega 2 +2}$ has cofinality $\aleph_{\omega+1}$. A proof of the conjecture below $\mathbf\delta^1_5=\aleph_{\omega^{\omega^\omega}+1}$ can be found in Jackson-Khafizov, Descriptions and cardinals below $\mathbf\delta^1_5$. Jackson's theory of descriptions can be used to verify the conjecture below $\aleph_{\epsilon_0}$, but a different approach would be needed for the general problem. Thanks to Yizheng Zhu for noticing the mistake in my first formulation of the question.)