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Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and Short extenders forcings II.

In Gitik's model the cardinal $\kappa=\sup(a)$ is a fixed point of the $\aleph$-function.

Question. Can we improve Gitik's result to get a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1$ such that $\kappa=\sup(a)$ is not a fixed point of the $\aleph$-function?

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    $\begingroup$ You should ask Moti directly. $\endgroup$
    – Asaf Karagila ♦
    Oct 29, 2015 at 7:11
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    $\begingroup$ That would be a huge advance for sure, and much more difficult! $\endgroup$
    – Todd Eisworth
    Oct 29, 2015 at 15:17
  • $\begingroup$ Mohammed, do you know off hand if the cardinal involved can be the least fixed point? $\endgroup$
    – Todd Eisworth
    Oct 30, 2015 at 14:58
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    $\begingroup$ @ToddEisworth No, but a few years ago I asked Gitik a related question and he said he's trying to push down the cardinal, but there are many difficulties in doing it. I don't know if he has made any progress on the problem since that time. $\endgroup$
    – Mohammad Golshani
    Oct 31, 2015 at 4:09

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