Hyper-inaccessible cardinals

Question

in definition of $\alpha$-inaccessible cardinals on Wikipedia, we can read :

For example, denote by $ψ_0(λ)$ the λth inaccessible cardinal, then the fixed points of $ψ_0$ are the 1-inaccessible cardinals.

1. But, the function $\psi_0$ is not a normal function (as it would implies that $cf(\psi_0(\omega))=\omega$, and, not so inaccessible !). So what (smallest, if any) hypothesis do we need to prove that $\psi_0$ has a fixed point and how to ?

2. same question for $\alpha$-hyper-inaccessible.

3. Is it sufficient for a cardinal to be inaccessible, hyper-inaccessible, hyper-hyper-inaccessible,... and so on, to be Mahlo ?

Answer 1

You don't need the normality in order to prove that $1$-inaccessible are the fixed points. It's in fact trivial from the fact that inaccessible cardinals are regular.

• $\kappa$ is $1$-inaccessible if and only if
• $\kappa$ is a limit of inaccessible cardinals if and only if
• $\kappa$ has $\kappa$ inaccessible below it if and only if
• $\kappa$ is the $\kappa$-th inaccessible.

The same point goes for the second question, there you can replace $\psi_0$ by $\psi_\alpha$ to argue for $\alpha+1$. For limit ordinals it follows from the definition, too.

Finally, you can find the answer to the last question here: mahlo and hyper-inaccessible cardinals