The Cardinality of the Continuum

This entry presents a short derivation of the cardinality of $\mathbb{R}$, namely that $|\mathbb{R}|=|2^{\mathbb{N}}|=2^{{\mathrm{\aleph }}_0}$. This is done by showing the injection $\mathbb{R}\to 2^{\mathbb{Q}},x\mapsto (-\mathrm{\infty },x)\cap \mathbb{Q}$ (i.e. Dedekind cuts) for one direction and the injection $2^{\mathbb{N}}\to \mathbb{Q},X\mapsto \sum _{n\in X}{3}^{-n}$, i.e. ternary fractions, for the other direction.