Cardinality of the Continuum¶

Definition¶

The cardinality of the continuum, often denoted $\mathfrak{c}$, is the cardinality of the real numbers:

Interpretation¶

$\mathfrak{c}$ is the size of the real line. It is an infinite size strictly larger than $\aleph_0$, the size of $\mathbb{N}$ (see aleph null): by Cantor's diagonal argument the real numbers cannot be listed as a sequence, so $\mathbb{R}$ is uncountable.

On the open interval $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ the graph of $\tan$ is continuous and increasing, with a vertical asymptote at each end, so every real height is reached exactly once. Hence $\tan\colon(-\tfrac{\pi}{2},\tfrac{\pi}{2})\to\mathbb{R}$ is a bijection, and this short interval has as many points as the whole real line: length plays no role in cardinality. In the same way $x\mapsto a+(b-a)x$ maps $(0,1)$ bijectively onto any open interval $(a,b)$ with $a<b$$b$.

Less obviously, $\mathfrak{c}=2^{\aleph_0}$$2^{{\aleph }_0}$, the cardinality of the power set of $\mathbb{N}$. Roughly, a number in $[0,1]$ has a binary expansion $0.b_1b_2b_3\ldots$, which corresponds to the set $\{n\colon b_n=1\}$ of positions with digit $1$. Even $\mathbb{R}^n$ has no more points than $\mathbb{R}$.

Between any two real numbers lies a rational number, yet $\mathbb{Q}$ only has cardinality $\aleph_0$. So the extra points are the irrational numbers: if they formed a countable set, $\mathbb{R}$ would be countable too. Whether some set has a size strictly between $\aleph_0$ and $\mathfrak{c}$ is the continuum hypothesis.

Key Results¶

• $\operatorname{card}(2^{\mathbb{N}})=2^{\aleph_0}=\operatorname{card}(\mathbb{R})=\mathfrak{c}$$\mathfrak{c}$.
• A bijection is the main tool for proving that two sets have the same cardinality.

Examples¶

• The interval $[0,1]$ has cardinality $\mathfrak{c}$.

Prerequisites¶

• Real Number

Related Concepts¶

• Cardinality
• Cantor's Diagonal Argument
• Aleph Null
• Cantor's Theorem
• Countable and Uncountable Sets

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: continuum cardinality, cardinality of the real numbers, beth one Keywords: cardinality of R Symbols: natural numbers, pi, rational numbers, real numbers

Last updated: 20 October 2026 03:02