Continuum hypothesis

There is no set whose cardinality is strictly between that of the integers $\mathbb{Z}$ and that of the real numbers $\mathbb{R}$; equivalently $2^{\aleph_0} = \aleph_1$.

Gödel constructed the constructible universe $L$ in 1938–40 to show CH is consistent with ZFC; Cohen invented forcing in 1963 to build models of ZFC + ¬CH, showing CH is also independent in the other direction. Together their work settles that CH cannot be decided from ZFC.