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Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous one. My question is:

Is $I$ the first fixed point of the function $\alpha \mapsto \aleph_\alpha$?

My background for this question is that while I was reading about ordinal collapsing functions (specifically about collapsing large cardinals) I found that by collapsing the least inaccessible cardinal ($I$) you got the first omega fixed point, which gave me the question, as if $I$ is the first fixed point then it is also a fixed point of the collapsing function, which wouldn’t be as useful.

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marked as duplicate by Asaf Karagila cardinals Jan 27 at 9:16

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    $\begingroup$ No. The first fixed point is the limit of $\aleph_0,\aleph_{\aleph_0},\dots$, which means it has cofinality $\aleph_0$ and is not regular. $\endgroup$ – Wojowu Jan 27 at 9:15
  • $\begingroup$ There are probably quite a few more of these around. $\endgroup$ – Asaf Karagila Jan 27 at 9:17
  • $\begingroup$ I tried looking for some, as I knew there would be them, but I couldn’t find any... $\endgroup$ – L. McDonald Jan 27 at 9:18