---
tag: 1RHTVMH34G
kind: concept
name: Weakly compact cardinal
msc: ["03E55", "03E02"]
specializes: [CJFEPPP9KA]
sources:
  - url: https://en.wikipedia.org/wiki/Weakly_compact_cardinal
same_as:
  wikidata: Q2905392
  wikipedia: Weakly compact cardinal
---

## Summary

A weakly compact cardinal is an uncountable cardinal $\kappa$ for which the infinite Ramsey theorem holds at $\kappa$.
This means that every colouring of the pairs of elements of $\kappa$ with two colours has a homogeneous set of size $\kappa$.
Weakly compact cardinals are [[inaccessible|NW4DREQ5DQ]] and even [[Mahlo|CJFEPPP9KA]].
The name comes from an equivalent property: a compactness theorem for the infinitary logic $\mathcal{L}_{\kappa, \kappa}$ in the case of theories with at most $\kappa$ sentences.
Every [[measurable cardinal|AKBWT323RQ]] is weakly compact.

## Explanations

### L1 human

Ramsey's theorem says that if one colours each pair of natural numbers red or blue, there is an infinite set of numbers whose pairs all have the same colour.
A weakly compact cardinal is an uncountable size of infinity for which the same is true.
For the first uncountable cardinal this fails, so a weakly compact cardinal must be very large.
It is a large cardinal that combines combinatorics, trees and logic in one notion.