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.
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 ?
same question for $\alpha$-hyper-inaccessible.
Is it sufficient for a cardinal to be inaccessible, hyper-inaccessible, hyper-hyper-inaccessible,... and so on, to be Mahlo ?