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?



Limit of Mahlo cardinals
There is some problems with definitions here.


1. **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.
2. 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.
3. 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.


