---
tag: 0WC28W9DNM
kind: object
name: Cardinality of the continuum
aliases: [continuum-cardinality]
msc: ["03E10"]
instance_of: [0EBACQFAZW]
uses: [0QKPNZKNEA, 0BC7KX51Z8, 0V2H5PGJ9B]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
    locator: lead section
same_as:
  wikidata: Q22952648
  wikipedia: Cardinality of the continuum
---

## Description

The cardinality of the continuum, written $\mathfrak{c}$, is the [[cardinal number|0QKPNZKNEA]] $2^{\aleph_0}$.
It is the cardinality of the [[power set|0BC7KX51Z8]] of $\mathbb{N}$, and of the set of all sequences of $0$ and $1$.
It is also the cardinality of $\mathbb{R}$. See [[the reals have the cardinality of the continuum|075PM9FH7A]].

## Remarks

By [[Cantor's theorem|0WTEMGYV2C]], $\mathfrak{c} > \aleph_0$.
Other sets of cardinality $\mathfrak{c}$ are each nonempty open interval of $\mathbb{R}$, the plane $\mathbb{R}^2$ and the set of irrational numbers.
The [[continuum hypothesis|0MN2MER9WN]] says that $\mathfrak{c} = \aleph_1$.

## Explanations

### L1 human

$\mathfrak{c}$ is the size of the real line.
It is a larger infinity than $\aleph_0$.
Surprisingly, a short interval, a whole line and a whole plane all have this same size.