The Cardinality of the Continuum

This entry presents a short derivation of the cardinality of $\mathbb{R}$, namely that $|\mathbb{R}| = |2^{\mathbb{N}}| = 2^{\aleph_0}$. This is done by showing the injection $\mathbb{R}\to 2^{\mathbb{Q}},\ x \mapsto (-\infty, x)\cap\mathbb{Q}$ (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.