Inaccessible and Mahlo Cardinals

Large cardinal axioms assert the existence of cardinals with extraordinary closure properties. They form a hierarchy measuring the consistency strength of mathematical theories.

Inaccessible Cardinals

Definition8.1Strongly inaccessible cardinal

A cardinal $\kappa > \aleph_0$ is strongly inaccessible if:

$\kappa$ is regular: $\mathrm{cf}(\kappa) = \kappa$ .
$\kappa$ is a strong limit: $2^\lambda < \kappa$ for all $\lambda < \kappa$ .

Equivalently, $\kappa$ is uncountable, regular, and $\beth_\alpha < \kappa$ for all $\alpha < \kappa$ .

Theorem8.1Inaccessible cardinals and models of ZFC

If $\kappa$ is strongly inaccessible, then $(V_\kappa, \in)$ is a model of ZFC. In particular, ZFC $+$ "there exists an inaccessible cardinal" proves Con(ZFC). By Godel's second incompleteness theorem, the existence of inaccessible cardinals is not provable in ZFC (if ZFC is consistent).

Mahlo Cardinals

Definition8.2Mahlo cardinal

A cardinal $\kappa$ is Mahlo if $\kappa$ is strongly inaccessible and the set of inaccessible cardinals below $\kappa$ is stationary in $\kappa$ .

More generally, define the Mahlo hierarchy: $\kappa$ is $0$ -Mahlo if inaccessible; $\kappa$ is $(\alpha+1)$ -Mahlo if the set of $\alpha$ -Mahlo cardinals below $\kappa$ is stationary; $\kappa$ is $\lambda$ -Mahlo (limit $\lambda$ ) if $\alpha$ -Mahlo for all $\alpha < \lambda$ . A cardinal is hyper-Mahlo if it is $\alpha$ -Mahlo for all $\alpha < \kappa$ .

ExampleThe lower large cardinal hierarchy

In increasing consistency strength:

$\text{inaccessible} < \text{Mahlo} < \text{weakly compact} < \text{indescribable} < \text{subtle} < \text{ineffable}$

Each level strictly exceeds the previous: a Mahlo cardinal is a limit of inaccessible cardinals, a weakly compact cardinal is a limit of Mahlo cardinals, etc.

Weakly Compact Cardinals

Definition8.3Weakly compact cardinal

A cardinal $\kappa$ is weakly compact if it is uncountable and has the tree property: every tree of height $\kappa$ in which every level has size $< \kappa$ has a branch of length $\kappa$ .

Equivalently (among many characterizations): $\kappa$ is inaccessible and satisfies $\kappa \to (\kappa)^2_2$ (the partition property: every 2-coloring of pairs from $\kappa$ has a monochromatic set of size $\kappa$ ).

RemarkWhy large cardinals matter

Large cardinals serve multiple purposes:

Consistency strength: They calibrate the strength of mathematical statements (e.g., projective determinacy is equiconsistent with infinitely many Woodin cardinals).
Regularity properties: Large cardinals imply that "definable" sets of reals are well-behaved (measurable, have the Baire property, etc.).
Inner models: The study of canonical inner models for large cardinals is central to modern set theory.