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Using the properties of the least inaccessible cardinal (being a regular fixed point of the aleph function) I was able to prove that the least inaccessible $\kappa$ is the $\kappa$-th fixed point of the aleph function which is monotonous and continuous at limit ordinals.

Is the converse true? Is the first cardinal with this property inaccessible?

Ajin Shaji Jose
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Arianit Niti Gashi
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  • If $\alpha$ is worldly (i.e., if $V_\alpha$ models $\mathsf{ZFC}$ then it is an $\alpha$th fixed point of aleph function but the least worldly cardinal has cofinality $\omega$. – Hanul Jeon Jun 22 '24 at 13:26
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    @Hanul: Even more can be said. The fixed points form a club, so ZF already proves the existence of fixed-fixed points, etc. – Asaf Karagila Jun 22 '24 at 17:04
  • @Asaf Thank you for your clarification. I vaguely noticed it after typing my comment, but I was too sleepy to comment it... – Hanul Jeon Jun 22 '24 at 23:49

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