>[!info] Definition > A cardinal $\kappa$ is *Mahlo* if and only if it is [[Inaccessible|(strongly) inaccessible]] and the [[Cardinal#Regular and singular cardinals|regular]] cardinals below $\kappa$ form a [[Club sets and stationary sets|stationary]] subset of $\kappa$. ^17f6e4 Equivalently, $\kappa$ is Mahlo if it is [[Cardinal#Regular and singular cardinals|regular]] and the [[Inaccessible|inaccessible]] cardinals below $\kappa$ are stationary. - Every Mahlo cardinal $\kappa$ is inaccessible, and indeed hyper-inaccessible and hyper-hyper-inaccessible, up to degree $\kappa$, and a limit of such cardinals. - If $\kappa$ is Mahlo, then it is Mahlo in any inner model, since the concept of stationarity is similarly downward absolute. Mahlo cardinals belong to the oldest large cardinals together with inaccessible and measurable. ## Weakly Mahlo >[!info] Definition > A cardinal $\kappa$ is *weakly Mahlo* if it is [[Cardinal#Regular and singular cardinals|regular]] and the set of [[Cardinal#Regular and singular cardinals|regular]] cardinals below $\kappa$ is [[Club sets and stationary sets|stationary]] in $\kappa$. ^c85288 Note that a weakly Mahlo cardinal is weakly inaccessible. If $\kappa$ is a [[Beth#Strong limit cardinal|strong limit]] and hence also (stronly) [[Inaccessible|inaccessible]], this is equivalent to $\kappa$ being Mahlo, since the [[Beth#Strong limit cardinal|strong limit]] cardinals form a closed unbounded subset in any [[Inaccessible|inaccessible]] cardinal. In particular, under the [[Continuum#Continuum hypothesis|GCH]], a cardinal is weakly Mahlo if and only if it is Mahlo. But in general, the concepts can differ, since adding an enormous number of Cohen reals will preserve all weakly Mahlo cardinals, but can easily destroy strong limit cardinals. Thus, every Mahlo cardinal can be made weakly Mahlo but not Mahlo in a forcing extension in which the continuum is very large. Nevertheless, every weakly Mahlo cardinal is Mahlo in any inner model of the GCH. ## The Mahlo hierarchy, Hyper-Mahlo etc. A cardinal $\kappa$ is *$1$-Mahlo* if the set of Mahlo cardinals is stationary in $\kappa$. This is a strictly stronger notion than merely asserting that $\kappa$ is a Mahlo limit of Mahlo cardinals, since in fact every $1$-Mahlo cardinal is a limit of such Mahlo-limits-of-Mahlo cardinals. (So there is an entire hierarchy of limits-of-limits-of-Mahloness between the Mahlo cardinals and the $1$-Mahlo cardinals.) More generally, >[!info] Definition >$\kappa$ is *$\alpha$-Mahlo* if it is Mahlo and for each $\beta\lt\alpha$ the class of $\beta$-Mahlo cardinals is stationary in $\kappa$. The cardinal $\kappa$ is *hyper-Mahlo* if it is $\kappa$-Mahlo. One may proceed to define the concepts of $\alpha$-hyper${}^\beta$-Mahlo by iterating this concept, iterating the stationary limit concept. Similarly, [[Baldwin - Generalizing the Mahlo Hierarchy, with Applications to the Mitchell Models]] defines an operation that gives a "weakly-Mahlo hierarchy". All such levels are swamped by the [[weakly compact]] cardinals, which exhibit all the desired degrees of hyper-Mahloness and more: Meta-ordinal terms are terms like $Ω^α · β + Ω^γ · δ +· · ·+Ω^\epsilon · \zeta + \theta$ where $α, β...$ are ordinals. They are ordered as if $Ω$ were an ordinal greater then all the others. $(Ω · α + β)$-Mahlo denotes $β$-hyper${}^α$-Mahlo, $Ω^2$-Mahlo denotes hyper${}^\kappa$-Mahlo $\kappa$ etc. Every weakly compact cardinal $\kappa$ is $\Omega^α$-Mahlo for all $α<\kappa$ and probably more. Similar hierarchy exists for [[Inaccessible|inaccessible]] cardinals below Mahlo. All such properties can be killed softly by forcing to make them any weaker properties from this family.{% cite Carmody2015 %} ## $\Sigma_n$-Mahlo etc. A regular cardinal $κ$ is $Σ_n$-Mahlo (resp. $Π_n$-Mahlo) if every club in $κ$ that is $Σ_n$-definable (resp. $Π_n$-definable) in $H(κ)$ contains an inaccessible cardinal. A regular cardinal $κ$ is $Σ_ω$-Mahlo if every club subset of $κ$ that is definable (with parameters) in $H(κ)$ contains an inaccessible cardinal. Every $Π_1$-Mahlo cardinal is an inaccessible limit of inaccessible cardinals. For Mahlo $κ$, the set of $Σ_ω$-Mahlo cardinals is stationary on $κ$. In {% cite Bosch2006 %} it is shown that every $Σ_ω$-[[weakly compact]] cardinal is $Σ_ω$-Mahlo and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is $Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of $Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary. These properties are connected with some forms of absoluteness. For example, the existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with the [[generic absoluteness axiom]] $\mathcal{A}(L(\mathbb{R}), Σ_ω , Γ ∩ absolutely−ccc)$ where $Γ$ is\the class of projective posets. This section from{% cite Bagaria2004 Bagaria2006 %}