---
tag: PGJNYGAN75
kind: definition
name: Mahlo cardinal
msc: ["03E55"]
defines: CJFEPPP9KA
uses: [NW4DREQ5DQ, YE6WZR46RQ, 13ZV2F8JAR]
sources:
  - url: https://en.wikipedia.org/wiki/Mahlo_cardinal
---

## Definition

Let $\kappa$ be an [[inaccessible cardinal|NW4DREQ5DQ]].
The cardinal $\kappa$ is *Mahlo* if the set
$$\set{\lambda < \kappa : \lambda \text{ is a regular cardinal}}$$
is a [[stationary subset|YE6WZR46RQ]] of $\kappa$, that is, it meets every closed unbounded subset of $\kappa$.

A weakly inaccessible cardinal $\kappa$ with the same property is *weakly Mahlo*.

## Remarks

For inaccessible $\kappa$, the uncountable strong limit cardinals below $\kappa$ form a closed unbounded set.
So $\kappa$ is Mahlo if and only if the inaccessible cardinals below $\kappa$ form a stationary set.
The least inaccessible cardinal $\kappa$ is not Mahlo.
Indeed, a regular cardinal among the uncountable strong limit cardinals below $\kappa$ would be an inaccessible cardinal below $\kappa$.
So the set of regular cardinals below $\kappa$ misses this closed unbounded set.
If $\kappa$ is Mahlo, then $\kappa$ is the $\kappa$-th inaccessible cardinal.
Every weakly compact cardinal is Mahlo.