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\title{The Cardinality of the Continuum}
\author{Manuel Eberl}
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\begin{abstract}
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.
\end{abstract}


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