Continuum hypothesis

Statement

There is no set $S$ with $\aleph_0 < |S| < 2^{\aleph_0}$. Equivalently, with the axiom of choice, $2^{\aleph_0} = \aleph_1$.

Remarks

In words: each infinite subset of $\mathbb{R}$ is countable or equinumerous with $\mathbb{R}$. The hypothesis is independent of ZFC: if ZFC is consistent, then ZFC proves neither the hypothesis nor its negation. The generalized continuum hypothesis says that $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for each ordinal $\alpha$.

History

Georg Cantor proposed the hypothesis in 1878. It was the first of the problems that David Hilbert presented in 1900. Kurt Gödel showed in 1940 that ZFC does not refute it, and Paul Cohen showed in 1963 that ZFC does not prove it.

Explanations

L1 human

Cantor showed that $\mathbb{R}$ is larger than $\mathbb{N}$. The continuum hypothesis says that no size lies strictly between them. The usual axioms of set theory cannot decide whether this is true.