A Mahlo cardinal (or strongly Mahlo cardinal) is an inaccessible cardinal \(\kappa\) such that the set of inaccessible cardinals below \(\kappa\) is a stationary subset of \(\kappa\) — that is, every closed unbounded set in \(\kappa\) 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 "\(\kappa\) is a Mahlo cardinal" may be rephrased as "\(\kappa\) is Mahlo," for example.)
If we weaken both instances of "inaccessible" in this definition to merely "regular," we get the weakly Mahlo cardinals. The two definitions are equivalent if the generalized continuum hypothesis is taken to be true; furthermore, if \(\kappa\) is weakly Mahlo and a strong limit cardinal, it is strongly Mahlo.
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. To define Mahlo cardinals we must additionally define the notion of a stationary set, and before we define that we need to define that of a club set.
We say that \(S\) is a club set in a 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 a limit ordinal \(\alpha\) iff \(S\) intersects all the club sets of \(\alpha\). Often when discussing stationary sets we only consider them to be well defined if \(\alpha\) has uncountable cofinality, as the structure of club sets in an ordinal of countable cofinality is much different (for example, there are disjoint club sets in this case) from an ordinal of uncountable cofinality.
Collapse[]
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 iff 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 iff 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 iff it is \(\kappa\)-Mahlo.
It is possible to define a notion of "Mahlo rank" in which \(\kappa\) can be \(>\kappa\)-Mahlo, by defining a well quasi-order \(<_{M}\) on the stationary subsets of \(\kappa\) which consist only regular cardinals, so that \(X <_{M} Y\) when \(Y \backslash \{\alpha \in X : X \cap \alpha\textrm{ is stationary in }\alpha\}\) is nonstationary (i.e. disjoint from some club set of \(\kappa\). Defining the Mahlo rank of \(\kappa\) to be the height of this order, and saying \(\kappa\) is \(\alpha\)-Mahlo when is rank is at least \(\alpha\), it is possible to define greatly Mahlo cardinals as the cardinals \(\kappa\) which are \(\kappa^{+}\)-Mahlo. All weakly compact cardinals are greatly Mahlo. It is possible to show that no cardinal \(\kappa\) is \(\left(2^{\kappa}\right)^{+}\)-Mahlo, and thus interestingly sufficiently high Mahlo rank relative to \(\kappa\) requires progressively stronger failure of the generalized continuum hypothesis: were to be \(\kappa^{++}\)-Mahlo then \(2^{\kappa} > \kappa^{+}\).
See also[]
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's \(\theta\) function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's \(\psi\) function · Rathjen's \(\Psi\) function · Stegert's \(\Psi\) function · Arai's \(\psi\) function
Uncountable cardinals: \(\omega_1\) · \(\omega\) fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)