Kerodon

Remark 4.7.2.13 (The Continuum Hypothesis). Let $\mathbf{R}$ be the set of real numbers. Then $| \mathbf{R} |$ is an uncountable cardinal (it is also the cardinality of the power set $P(\operatorname{\mathbf{Z}})$). The continuum hypothesis is the assertion that $| \mathbf{R} |$ coincides with the smallest uncountable cardinal $\aleph _1$. This was a central question in the early days of set theory (and first of Hilbert's celebrated list of problems for the mathematics of the 20th century). It is now known to be neither provable nor disprovable from the axioms of Zermelo-Fraenkel set theory (assuming that they are consistent), thanks to the work of Gödel ([MR0002514]) and Cohen ([MR157890], [MR159745]).