---
tag: 13ZV2F8JAR
kind: concept
name: Regular cardinal
msc: ["03E10", "03E04"]
specializes: [0EBACQFAZW]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_cardinal
same_as:
  wikidata: Q1193137
  wikipedia: Regular cardinal
---

## Summary

A regular cardinal is an infinite [[cardinal|0EBACQFAZW]] that cannot be reached by fewer smaller steps than itself: its [[cofinality|BAWHE1W6MH]] equals itself.
An infinite cardinal that is not regular is singular.
Regular cardinals are the natural setting for clubs, stationary sets and much of infinite combinatorics, and the [[precise definition|ZCE333FAWC]] is short.

## Explanations

### L1 human

The cardinal $\aleph_0$ is regular: a finite union of finite sets is finite, so finitely many finite steps never reach $\omega$.
The cardinal $\aleph_1$ is regular too, because a countable union of countable sets is countable.
But $\aleph_\omega$ is singular: the countably many smaller cardinals $\aleph_0, \aleph_1, \aleph_2, \ldots$ already climb up to it.
So regularity says a cardinal is not the limit of a short sequence of smaller cardinals.