---
tag: NW4DREQ5DQ
kind: concept
name: Inaccessible cardinal
msc: ["03E55"]
specializes: [X1FG7PSCTY]
sources:
  - url: https://en.wikipedia.org/wiki/Inaccessible_cardinal
same_as:
  wikidata: Q1331373
  wikipedia: Inaccessible cardinal
---

## Summary

An inaccessible cardinal is an uncountable [[cardinal|0EBACQFAZW]] $\kappa$ that cannot be reached from smaller cardinals.
Power sets of smaller sets, and unions of fewer than $\kappa$ smaller sets, stay smaller than $\kappa$.
It is the smallest kind of [[large cardinal|X1FG7PSCTY]].
If $\kappa$ is inaccessible, the sets of rank less than $\kappa$ form a model of ZFC.
So ZFC cannot prove that inaccessible cardinals exist, if ZFC is consistent.
In category theory, inaccessible cardinals give [[Grothendieck universes|SMMK9E0MTM]].

## Explanations

### L1 human

The natural numbers cannot be reached from finite numbers: sums, products and powers of finite numbers stay finite.
An inaccessible cardinal is an uncountable size with the same property.
Taking power sets of smaller sets, or unions of fewer than $\kappa$ sets each smaller than $\kappa$, never gives a set of size $\kappa$.
So everything built from below stays below, and the sets below $\kappa$ form a whole universe of their own.