---
tag: 0518MM6BE1
kind: statement
role: theorem
name: The real numbers are uncountable
aliases: [uncountability-of-the-reals]
msc: ["03E10"]
uses: [0X55M9XET4]
sources:
  - url: https://en.wikipedia.org/wiki/Cantor's_first_set_theory_article
    locator: lead section
---

## Statement

The set $\mathbb{R}$ of real numbers is [[uncountable|0X55M9XET4]].

## Remarks

Cantor proved it in his article of 1874. That proof uses nested intervals.
A later proof uses [[Cantor's diagonal argument|0XCH8QVDCQ]] on decimal or binary expansions.
Since the algebraic numbers are countable, the result shows that transcendental numbers exist. Cantor gave this application in the same article.

## Explanations

### L1 human

Each list of real numbers misses some real number.
So there are more real numbers than natural numbers, although both sets are infinite.