---
tag: BAWHE1W6MH
kind: concept
name: Cofinality
msc: ["03E04", "03E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Cofinality
same_as:
  wikidata: Q1283623
  wikipedia: Cofinality
---

## Summary

The cofinality of a limit [[ordinal|0YSCSX5YZC]] is the least length of a sequence of smaller ordinals that climbs all the way up to it.
The [[precise definition|NTE5C260S1]] uses cofinal subsets, and it applies also to partially ordered sets.
Cofinality splits infinite cardinals into [[regular|13ZV2F8JAR]] and singular ones, and it constrains the possible values of cardinal exponentiation.

## Explanations

### L1 human

To reach the top of $\omega$, the natural numbers, you need infinitely many steps $0, 1, 2, \ldots$, so its cofinality is $\omega$.
The much bigger cardinal $\aleph_\omega$ can also be reached in $\omega$ steps, through $\aleph_0, \aleph_1, \aleph_2, \ldots$, so its cofinality is also $\omega$.
In contrast, $\aleph_1$ cannot be reached by any countable sequence of countable ordinals, since a countable union of countable sets is countable.
So the cofinality of $\aleph_1$ is $\aleph_1$ itself.
Cofinality measures how hard it is to climb to the top.