---
tag: 0X55M9XET4
kind: definition
name: Uncountable set
msc: ["03E10"]
defines: 0W7FBYS1F1
uses: [0ZG61TFBES]
sources:
  - url: https://en.wikipedia.org/wiki/Uncountable_set
    locator: lead section
---

## Definition

A [[set|0SJ3RX9TXA]] $A$ is uncountable when it is not [[countable|0ZG61TFBES]], that is, when there is no [[injective function|0PVG1CRAJN]] from $A$ to $\mathbb{N}$.

## Remarks

The real numbers and the [[power set|0BC7KX51Z8]] of $\mathbb{N}$ are uncountable.
With the axiom of choice, a set is uncountable exactly when its cardinality is at least $\aleph_1$, the least cardinal larger than $\aleph_0$.
A superset of an uncountable set is uncountable.

## Explanations

### L1 human

For an uncountable set, every attempt to list its elements misses some of them.
[[Cantor's diagonal argument|0XCH8QVDCQ]] shows how to find a missing element.