Limit of Mahlo cardinals What cardinal is the limit of this fundamental sequence?
{The first Mahlo cardinal, the first 1-Mahlo cardinal, the first hyper-Mahlo cardinal, the first hyper-hyper-Mahlo cardinal, the first hyper-hyper-hyper-Mahlo cardinal, ...}
where the definition of a $\alpha$-Mahlo cardinal is as follows:
If we define a function where $\psi_0(\lambda)$ is the $\lambda$th ($\alpha$-1)-Mahlo cardinal, then the fixed points of $\psi_0$ are the α-Mahlo cardinals, and the 0-Mahlo cardinals are just the Mahlo cardinals.
Thanks!
EDIT: Based on comments, I have a new easier to understand question:
What is the name of the first cardinal that has the properties of being Mahlo, hyper-Mahlo, hyper-hyper-Mahlo, hyper-hyper-hyper-Mahlo, hyper-hyper-hyper-hyper-Mahlo, and so on?
 A: There is some problems with definitions here.

*

*The definition you use of $\alpha+1$-Mahlo cardinals is a cardinal $\kappa$ such that $\kappa$ is the $\kappa$th $\alpha$-Mahlo cardinal.
This is not the standard definition. Usually, a cardinal is said to be $\alpha+1$-Mahlo if $\{\beta\lt\kappa|\beta\text{ is }\alpha\text{-Mahlo}\}$ is stationary.
We will call the first notion $\alpha$-Mahlo, and the second notion $\alpha$-Mahlo* (You will never find that notation in literature, I just wanted to clarify which definition I am using)
If $\kappa$ is $\alpha$-Mahlo* for $\alpha\gt 0$, then $\kappa$ is $\alpha$-Mahlo, $\alpha+1$-Mahlo, $\kappa$-Mahlo, and so on.


*If we say that $\kappa$ is hyper-Mahlo if $\kappa$ is $\kappa$-Mahlo, and then define $\alpha+1$-hyper-Mahlo cardinals as fixed points of $\alpha$-hyper-Mahlo cardinals, hyper$^2$-Mahlo as $\kappa$-hyper-Mahlo cardinals, and so on, then we can get the notion of $\alpha$-hyper$^n$-Mahlo cardinals.
I should stress that this is not the standard definition, but it appears to be the one you are using.
Normally, we say that $\kappa$ is hyper-Mahlo* if $\kappa$ is $\kappa$-Mahlo*, and then define $\alpha+1$-hyper-Mahlo* cardinals as stationary limits of $\alpha$-hyper-Mahlo* cardinals, hyper$^2$-Mahlo* as $\kappa$-hyper-Mahlo cardinals*, and so on.
If $\kappa$ is $1$-Mahlo*, then $\kappa$ is hyper-Mahlo, hyper$^2$-Mahlo, hyper$^3$-Mahlo, and so on.


*The first cardinal $\kappa$ that is hyper$^n$-Mahlo for every $n\lt\omega$, is not the same as  the cardinal $\kappa'=\text{sup}\{\text{The least Mahlo cardinal, the least hyper-Mahlo cardinal, the least hyper}^2\text{-Mahlo cardinal...}\}$
$\kappa'$ has cofinality $\omega$, and so is not even inaccessible, and  $\kappa'\lt\kappa$.
There is no official name for $\kappa'$. You could call $\kappa$ the least hyper$^\omega$-Mahlo cardinal, but don't forget that the definition you are using of hyper$^n$-Mahlo cardinals is not the standard one! If you use the standard definition, using stationary limits instead of fixed points, then $\kappa$ would be the least hyper$^\omega$-Mahlo cardinal.
