Continuum Hypothesis

Definition

$\nexists S:{\aleph }_0<|S|<2^{{\aleph }_0}$

There is no set whose cardinality is strictly between that of the $\mathbb{Z}$^[${\aleph }_0$], and $\mathbb{R}$^[$2^{{\aleph }_0}$]

The Generalized Continuum Hypothesis

$\forall \kappa ,\nexists \lambda :\kappa <\lambda <2^{\kappa }$ where $k$ and $\lambda$ are the Cardinal Number of infinite set

Facts

The continuum hypothesis is independent of ZFC Set Theory