Cardinality of the continuum

Description

The cardinality of the continuum, written $\mathfrak{c}$, is the cardinal number $2^{\aleph_0}$. It is the cardinality of the power set 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.

Remarks

By Cantor's theorem, $\mathfrak{c} > \aleph_0$. 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 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.