---
tag: CJFEPPP9KA
kind: concept
name: Mahlo cardinal
msc: ["03E55"]
specializes: [NW4DREQ5DQ]
sources:
  - url: https://en.wikipedia.org/wiki/Mahlo_cardinal
same_as:
  wikidata: Q3277546
  wikipedia: Mahlo cardinal
---

## Summary

A Mahlo cardinal is an [[inaccessible cardinal|NW4DREQ5DQ]] $\kappa$ such that the inaccessible cardinals below $\kappa$ form a [[stationary set|YE6WZR46RQ]] in $\kappa$.
So a Mahlo cardinal has very many inaccessible cardinals below it, and it is the $\kappa$-th inaccessible cardinal.
The notion is named after Paul Mahlo.
Mahlo cardinals are stronger than inaccessible cardinals, and weaker than [[weakly compact cardinals|1RHTVMH34G]].

## Explanations

### L1 human

An inaccessible cardinal is a size of infinity that cannot be reached from below.
A Mahlo cardinal is an inaccessible cardinal that has inaccessible cardinals below it in a strong sense: any closed unbounded set of smaller ordinals contains one of them.
The first inaccessible cardinal is not Mahlo, since there is no inaccessible cardinal below it at all.
So Mahlo cardinals are a step up in the hierarchy of large cardinals.