# Singularity of small cardinals under AD

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 $\omega_\omega$.

• (By the way, I would not call these results "folklore". It is well documented who proved them, where, and how.) – Andrés E. Caicedo Jun 9 '15 at 22:04
• @AndresCaicedo I couldn't have found a reference as for who proved these, so I thought it's a folklore result (which, I now see, it's not). Feel free to repost your comment as an answer, so I can accept it. – Wojowu Jun 10 '15 at 4:41

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.)

• Thanks a lot for a detailed reference. – Wojowu Jun 10 '15 at 5:05
• So the open problem is to get singular successor cardinals of cofinality $<\omega_2$ below $\Theta$? – Asaf Karagila Jun 10 '15 at 17:37
• $\aleph_{\omega+1}$ is regular, @Asaf. There are many regular successor and many singular successor cardinals below $\Theta$. – Andrés E. Caicedo Jun 10 '15 at 17:44
• These structure consequences of $\sf AD$ always feel funky to me. One day I'll get the hang of it... – Asaf Karagila Jun 10 '15 at 17:50
• After your edit, I'm not so sure about my previous comment anymore! :-) – Asaf Karagila Jun 12 '15 at 15:27