The real numbers are uncountable

Statement

The set $\mathbb{R}$ of real numbers is uncountable.

Remarks

Cantor proved it in his article of 1874. That proof uses nested intervals. A later proof uses Cantor's diagonal argument on decimal or binary expansions. Since the algebraic numbers are countable, the result shows that transcendental numbers exist. Cantor gave this application in the same article.

Explanations

L1 human

Each list of real numbers misses some real number. So there are more real numbers than natural numbers, although both sets are infinite.