Mahlo cardinals are a type of large cardinal κ such that κ is both inaccessible and the set of weak or strong inaccessibles beneath them is stationary within them. An ordered set α is said to be stationary in κ if α intersects all the closed unbounded subsets β of κ (sets cofinal to κ and for which all the limit points of sequences of cardinality less than κ are contained in β).
Discovery[]
Weakly inaccessible cardinals were formulated in 1908 by Felix Hausdorff under the name inaccessibles (unerreichbare or 'unreachable' cardinals). At this time they were just called "inaccessibles," and so too weakly Mahlos (weakly inaccessible κ such that the set of weakly inaccessibles beneath them are stationary in κ) were just called "Mahlo". They were discovered in 1911 by Paul Mahlo. In modern times the nomenclature is the other way around and "inaccessible" actually refers to the strong inaccessibles (discovered in 1930 by Sierpinski, Tarski, and Zermelo) while "Mahlo" refers to the strongly Mahlo cardinals (strongly inaccessible κ such that the set of strongly inaccessibles beneath them are stationary in κ).
Definition[]
An equivalent definition of Mahlo cardinals to the one given at the outset is a set κ such that for all normal ordinal functions φ and elements α of κ there exists a regular fixed point of α ↦ φ(α) also within κ. Stating "all normal ordinal functions have a regular fixed point" (termed Axiom F) does not imply the existence of a Mahlo set but only says the universe is Mahlo (i.e. VM ⊨ ZFC + 「Axiom F」).
The reason that implication of the existence of regular fixed points for all normal functions is related to Mahlos is because the inaccessibles can be built up using iterated regular fixed points (i.e. imagine a Veblen-type hierarchy φ(αβ,...,α2,α1) where φ(α) is the (1 + α)th fixed point of β ↦ ρ(β) where ρ enumerates regular cardinals). Axiom F diagonalizes over this by quantifying over functions (a second-order logic operation) and thereby inaccessibility as a whole. The 1st Mahlo cardinal is therefore a limit of 0-inaccessibles (φ(α)), 1-inaccessibles (φ(1,α)), ω-inaccessibles (φ(ω,α)), hyper-inaccessibles (φ(1,0,α)), hyper-hyper-inaccessibles (φ(1,1,α)), super-hyper-inaccessibles (φ(2,0,α)), and so forth.
n-Mahlo and hyper-Mahlo cardinals[]
In general, a cardinal κ is (α + 1)-Mahlo if the set of α-Mahlo cardinals less than κ is stationary in κ. For a limit ordinal α, κ is α-Mahlo if κ is β-Mahlo for all β < α. Naturally this can be extended in the same way inaccessible hierarchies can, so that hyper-Mahlos are defined as κ which are κ-Mahlo, hyper-hyper-Mahlos as κ which are κ-hyper-Mahlo, and so forth.
Because Mahlos are defined using generalization over regular fixed points they are useful in OCFs where they can be collapsed into inaccessibles and hyper-inaccessibles, which are like the bridge to Mahlos. In general large cardinal axioms do not have bridges between them as conducive to OCF analysis as this (i.e. the collapse of 0# implying cardinals or rank-into-rank cardinals is likely unknown).