Mahlo cardinal
A cardinal $\kappa$ is Mahlo if and only if it is inaccessible and the regular cardinals below $\kappa$ form a stationary subset of $\kappa$. Equivalently, $\kappa$ is Mahlo if it is regular and the 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.
Ord is Mahlo is a scheme asserting that regular cardinals form a stationary class. It is weaker than the existence of a Mahlo cardinal (or even pseudo $0$-uplifting cardinal[1]).
Mahlo cardinals belong to the oldest large cardinals together with inaccessible and measurable. Please add more history.
Weakly Mahlo
A cardinal $\kappa$ is weakly Mahlo if it is regular and the set of regular cardinals below $\kappa$ is stationary in $\kappa$. If $\kappa$ is a strong limit and hence also inaccessible, this is equivalent to $\kappa$ being Mahlo, since the strong limit cardinals form a closed unbounded subset in any inaccessible cardinal. In particular, under the 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.
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, $\kappa$ 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. 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 cardinals below Mahlo. All such properties can be killed softly by forcing to make them any weaker properties from this family.[2]
$\Sigma_n$-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 [3] 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 axioms of generic absoluteness. For example:
- There is a, provably in ZFC, ccc poset $\mathbb{P}$ which is both $Σ_2$ and $Π_2$ definable in $H(ω_1)$, without parameters, and for which the axiom $\mathcal{A}(H(ω_1), Σ_3, \mathbb{P})$ fails if $ω_1$ is not a $Π_1$-Mahlo cardinal in $L$.
- The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with both $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω}, Γ)$ and $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.
- The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of both $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω}, Γ)$ and $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.
References
- Hamkins, Joel David and Johnstone, Thomas A. Resurrection axioms and uplifting cardinals. , 2014. www arχiv bibtex
- Carmody, Erin Kathryn. Force to change large cardinal strength. , 2015. www arχiv bibtex
- Bosch, Roger. Small Definably-large Cardinals. Set Theory Trends in Mathematics pp. 55-82, 2006. DOI bibtex
- Bagaria, Joan and Bosch, Roger. Proper forcing extensions and Solovay models. Archive for Mathematical Logic , 2004. www DOI bibtex
- Bagaria, Joan. Axioms of generic absoluteness. Logic Colloquium 2002 , 2006. www DOI bibtex