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I've been reading about set theory and the difference between small and large cardinals. since taking the power set of small cardinals (alephs) allows us to create larger cardinals/alephs I know that an inaccessible cardinal is equivalent to the regular aleph fixed point, and cannot be reached by taking the power sets of alephs. So my questions are:

  • what would the power set of an inaccessible cardinal be?
  • what is the regular fixed point of the first inaccessible cardinal? (basically, a cardinal that is larger than the first inaccessible cardinal to the same degree the first inaccessible cardinal is larger than the first small cardinals like aleph 0)
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    $\begingroup$ What is the power set of anything? It's just the set of all subsets of that thing. And what does the second question even mean (syntactically, I can't parse it, what is a "fixed point of a cardinal")? $\endgroup$
    – Asaf Karagila
    Jun 2, 2023 at 8:55
  • $\begingroup$ @Asaf Karagila: what is a "fixed point of a cardinal" -- Probably the intended meaning is what Joel David Hamkins describes in this mathoverflow question. I don't know much about them, but my (and others) comments to Veblen function with uncountable ordinals & beyond may be of help, as well as the mathoverflow questions Mahlo cardinal and hyper k-inaccessible cardinal AND Limit of Mahlo cardinals. $\endgroup$ Jun 2, 2023 at 9:14
  • $\begingroup$ @AsafKaragila i meant the aleph fixed point, something outlined in this thread math.stackexchange.com/questions/3089308/… would it not be an inaccessible cardinal? $\endgroup$
    – Adithya
    Jun 3, 2023 at 5:10
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    $\begingroup$ “A cardinal that’s larger than the first inaccessible to the same degree that the first inaccessible is larger than $\aleph_0$”… if that means anything I would think it refers to the second inaccessible. $\endgroup$ Jun 3, 2023 at 16:04
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    $\begingroup$ The second inaccessible is the second least inaccessible… the next one. It is “inaccessible” from the least inaccessible in exactly the same manner as the least inaccessible is “inaccessible” from $\aleph_0$. A 1-inaccessible (i.e. a fixed point in the enumeration of inaccessibles) is much larger. $\endgroup$ Jun 3, 2023 at 23:10

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