---
tag: 075PM9FH7A
kind: statement
role: theorem
name: The reals have the cardinality of the continuum
aliases: [cardinality-of-the-reals]
msc: ["03E10"]
uses: [0WC28W9DNM, 0A9V6RKQG6, 0GPFKS96T7]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
    locator: Cardinal equalities
---

## Statement

The set $\mathbb{R}$ of real numbers has [[cardinality|0JTGRV4WT4]] $2^{\aleph_0}$:
$$|\mathbb{R}| = \mathfrak{c} = |\mathcal{P}(\mathbb{N})|.$$

## Remarks

One proof uses the [[Schröder–Bernstein theorem|0A9V6RKQG6]]. A real number gives a set of rationals, the rationals less than it, so $|\mathbb{R}| \le |\mathcal{P}(\mathbb{Q})| = 2^{\aleph_0}$. Binary sequences give an injection in the other direction, for example through decimal expansions with digits $0$ and $1$.
With [[Cantor's theorem|0WTEMGYV2C]], this gives another proof that $\mathbb{R}$ is [[uncountable|0518MM6BE1]].

## Explanations

### L1 human

The real numbers and the subsets of $\mathbb{N}$ have the same size.
Roughly, a real number is an infinite sequence of digits, and an infinite sequence of $0$ and $1$ is a subset of $\mathbb{N}$.