---
tag: NTE5C260S1
kind: definition
name: Cofinality
msc: ["03E04", "03E10"]
defines: BAWHE1W6MH
uses: [0RDJK7EQYY, 011C29Q1S1, 0XHDTE5YMZ]
sources:
  - url: https://en.wikipedia.org/wiki/Cofinality
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A subset $A \subseteq P$ is cofinal in $P$ if for each $p \in P$ there is $a \in A$ with $p \le a$.
Let $\alpha$ be an [[ordinal|0RDJK7EQYY]], ordered by $\in$.
The cofinality $\operatorname{cf}(\alpha)$ of $\alpha$ is the least ordinal $\beta$ such that $\alpha$ has a cofinal subset of order type $\beta$.

## Remarks

Equivalently, $\operatorname{cf}(\alpha)$ is the least $\beta$ for which there is a [[function|0XHDTE5YMZ]] $f \colon \beta \to \alpha$ whose range is cofinal in $\alpha$.
Example: $\operatorname{cf}(\aleph_\omega) = \omega$.
Check: the set $\set{\aleph_n : n < \omega}$ has order type $\omega$ and is cofinal in $\aleph_\omega$, and no finite subset of a limit ordinal is cofinal.
Non-example of a cofinal subset: a finite subset $A$ of $\omega$ is not cofinal, since $\max A + 1$ has no element of $A$ above it.
For a successor ordinal $\alpha + 1$, the set $\set{\alpha}$ is cofinal, so $\operatorname{cf}(\alpha + 1) = 1$, and $\operatorname{cf}(0) = 0$.
Always $\operatorname{cf}(\operatorname{cf}(\alpha)) = \operatorname{cf}(\alpha)$, and the cofinality of a limit ordinal is a regular cardinal.
In ZFC, König's theorem gives $\operatorname{cf}(2^\kappa) > \kappa$ for each infinite cardinal $\kappa$.