Mahlo cardinal

Remarks

For inaccessible $\kappa$, the uncountable strong limit cardinals below $\kappa$ form a closed unbounded set. So $\kappa$ is Mahlo if and only if the inaccessible cardinals below $\kappa$ form a stationary set. The least inaccessible cardinal $\kappa$ is not Mahlo. Indeed, a regular cardinal among the uncountable strong limit cardinals below $\kappa$ would be an inaccessible cardinal below $\kappa$. So the set of regular cardinals below $\kappa$ misses this closed unbounded set. If $\kappa$ is Mahlo, then $\kappa$ is the $\kappa$-th inaccessible cardinal. Every weakly compact cardinal is Mahlo.