Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers ${\displaystyle \mathbb {R} }$, sometimes called the continuum. It is an infinite cardinal number and is denoted by ${\displaystyle {\mathfrak {c}}}$ (lowercase Fraktur "c") or ${\displaystyle |\mathbb {R} |}$.^[1]

The real numbers ${\displaystyle \mathbb {R} }$ are more numerous than the natural numbers ${\displaystyle \mathbb {N} }$. Moreover, ${\displaystyle \mathbb {R} }$ has the same number of elements as the power set of ${\displaystyle \mathbb {N} .}$ Symbolically, if the cardinality of ${\displaystyle \mathbb {N} }$ is denoted as ${\displaystyle \aleph _{0}}$, the cardinality of the continuum is

${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}>\aleph _{0}\,.}$

This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.

Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with ${\displaystyle \mathbb {R} .}$ This is also true for several other infinite sets, such as any n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (see space filling curve). That is,

${\displaystyle |(a,b)|=|\mathbb {R} |=|\mathbb {R} ^{n}|.}$

The smallest infinite cardinal number is ${\displaystyle \aleph _{0}}$ (aleph-null). The second smallest is ${\displaystyle \aleph _{1}}$ (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between ${\displaystyle \aleph _{0}}$ and ${\displaystyle {\mathfrak {c}}}$, means that ${\displaystyle {\mathfrak {c}}=\aleph _{1}}$.^[2] The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).