---
tag: X1FG7PSCTY
kind: concept
name: Large cardinal
msc: ["03E55"]
specializes: [0EBACQFAZW]
sources:
  - url: https://en.wikipedia.org/wiki/Large_cardinal
same_as:
  wikidata: Q1548262
  wikipedia: Large cardinal
---

## Summary

A large cardinal is an infinite [[cardinal number|0EBACQFAZW]] with a property that makes it very large.
If ZFC is consistent, ZFC cannot prove that such cardinals exist.
Examples are [[inaccessible|NW4DREQ5DQ]], [[Mahlo|CJFEPPP9KA]], [[weakly compact|1RHTVMH34G]], [[measurable|AKBWT323RQ]], [[Woodin|CMC2C1D53W]] and [[supercompact|91GE4C1BC5]] cardinals.
There is no precise definition of the term; it names a family of axioms that is, as far as is known, ordered by consistency strength.
Large cardinal axioms measure the strength of other statements, and they decide many questions about sets of reals, such as [[projective determinacy|2GY3131ZPS]].

## Explanations

### L1 human

The usual axioms of set theory give infinitely many sizes of infinity, one after another.
A large cardinal is a size of infinity so big that the usual axioms cannot reach it from below.
For example, an inaccessible cardinal cannot be reached by taking power sets or unions of fewer sets.
Adding a large cardinal as a new axiom gives a stronger theory, and set theorists compare statements by asking which large cardinal they need.