Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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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.

Mahlo cardinals belong to the oldest large cardinals together with
inaccessible and measurable. *Please add more history.*

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.

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.(Carmody, 2015)

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 (Bosch, 2006) 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(Bagaria & Bosch, 2004; Bagaria, 2006)

- Carmody, E. K. (2015).
*Force to change large cardinal strength*. https://academicworks.cuny.edu/gc_etds/879/ - Bosch, R. (2006). Small Definably-large Cardinals.
*Set Theory. Trends in Mathematics*, 55–82. https://doi.org/10.1007/3-7643-7692-9_3 - Bagaria, J., & Bosch, R. (2004). Proper forcing extensions and Solovay models.
*Archive for Mathematical Logic*. https://doi.org/10.1007/s00153-003-0210-2 - Bagaria, J. (2006). Axioms of generic absoluteness.
*Logic Colloquium 2002*. https://doi.org/10.1201/9781439865903