# What is the power set of an inaccessible cardinal?

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)
• 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")? Jun 2, 2023 at 8:55
• @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. Jun 2, 2023 at 9:14
• @AsafKaragila i meant the aleph fixed point, something outlined in this thread math.stackexchange.com/questions/3089308/… would it not be an inaccessible cardinal? Jun 3, 2023 at 5:10
• “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. Jun 3, 2023 at 16:04
• 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. Jun 3, 2023 at 23:10