A Mahlo cardinal (or strongly Mahlo cardinal) is an inaccessible cardinal \(\alpha\) such that the set of inaccessible cardinals below \(\alpha\) is a stationary subset of \(\alpha\) — that is, every closed unbounded set in \(\alpha\) contains an inaccessible cardinal (in which the Von Neumann definition of ordinals is used). The smallest Mahlo cardinal is sometimes called "the" Mahlo cardinal \(M\). (The eponym "Mahlo" has been appropriated as an adjective, so "\(\alpha\) is a Mahlo cardinal" may be rephrased as "\(\alpha\) is Mahlo," for example.)
If we weaken "inaccessible" to merely "regular," we get the weakly Mahlo cardinals. The two definitions are equivalent if the generalized continuum hypothesis is taken to be true.
Neither Mahlo nor weakly Mahlo cardinals can be proven to exist in ZFC (assuming it is consistent), not even if we assume the existence of any number of inaccessible cardinals (also assuming it is consistent). Nevertheless, it's believed that the existence of these cardinals is consistent with ZFC.
Club sets and stationary sets
The notions of regularity and inaccessibility are explained in the article for inaccessible cardinals. The Mahlo cardinal requires us to define in addition the idea of a stationary set, and before we define that we need to define a club set.
We say that \(S\) is a club set in limit ordinal \(\alpha\) iff \(S\) is a subset of \(\alpha\), \(S\) is closed in \(\alpha\), and \(S\) is unbounded in \(\alpha\). Intuitively, \(S\) is a part of \(\alpha\), \(S\) contains all its own limit points provided that they are less than \(\alpha\), and every element in \(\alpha\) is exceeded by some element in \(S\).
An example: all the countable limit ordinals form a club set (call it \(A\)) in \(\omega_1\). The limit of a set of countable limit ordinals is always a countable limit ordinal, with one exception — the limit of \(A\) is \(\omega_1\), and \(\omega_1\) is not in \(A\). But since \(\omega_1\) is not a member of \(\omega_1\) either, this does not matter. \(A\) is unbounded in \(\omega_1\), since any countable ordinal is beaten by a member in \(A\). Therefore, \(A\) is club in \(\omega_1\). (in general, it is easy to see that given a limit ordinal \(\alpha\), the set of ordinals below \(\alpha\) is club in \(\alpha\) because it coincides with \(\alpha\) itself.)
We say that \(S\) is a stationary set in limit ordinal \(\alpha\) iff \(S\) intersects all the club sets of \(\alpha\).
The Mahlo cardinals are most relevant to googology through ordinal collapsing functions such as Rathjen's ψ function, and ordinal notations associated to them. The fast-growing hierarchy along such notations has been used for comparisons of strength of functions, such as ones associated to ε function and KumaKuma ψ function.
Higher-order Mahlo cardinals
A cardinal \(\kappa\) is 1-Mahlo if it is Mahlo and the set of Mahlo cardinals less than \(\kappa\) is a stationary subset of \(\kappa\). In general, a cardinal \(\kappa\) is \(\alpha\)-Mahlo if it is Mahlo and, for all \(\beta < \alpha\), the set of \(\beta\)-Mahlo cardinals less than \(\kappa\) is a stationary subset of \(\kappa\).
A cardinal \(\kappa\) is hyper-Mahlo if it is \(\kappa\)-Mahlo.