---
tag: YE6WZR46RQ
kind: concept
name: Stationary set
msc: ["03E05", "03E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Stationary_set
same_as:
  wikidata: Q1528401
  wikipedia: Stationary set
---

## Summary

A stationary set in a regular uncountable cardinal $\kappa$ is a subset of $\kappa$ that meets every [[club set|YW8YSCFPG5]].
If club sets are the "almost all" sets, stationary sets are the sets that are "not negligible".
They are central in combinatorial set theory, through [[Fodor's lemma|55W1M20Q3F]], the [[diamond principle|HFQSSB1Z23]] and forcing axioms; the [[precise definition|6FDYJVG1CK]] is given separately.

## Explanations

### L1 human

Among the countable ordinals, the club sets are those that are "almost everything".
A stationary set is one that you cannot avoid with a club set: whichever club set you pick, the two share a point.
Every club set is stationary.
One level up, the set of all ordinals of cofinality $\omega$ below $\omega_2$ is stationary in $\omega_2$.
A bounded set, for instance all ordinals below $\omega \cdot 2$, is not stationary in $\omega_1$, because the ordinals above it form a club set that misses it.