The generalized continuum hypothesis states that for any infinite cardinal $\lambda$ there is no cardinal $\kappa$ such that $\lambda<\kappa<2^{{\lambda}}$.

An equivalent condition is that $\aleph_{{\alpha+1}}=2^{{\aleph_{\alpha}}}$ for every ordinal $\alpha$. Another equivalent condition is that $\aleph_{\alpha}=\beth_{\alpha}$ for every ordinal $\alpha$.

Like the continuum hypothesis, the generalized continuum hypothesis is known to be independent of the axioms of ZFC.