Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 5 / ii / Solution

The generalized continuum hypothesis isEquivalently, for every ordinal $\alpha$, $2^{{\aleph }_{\alpha }}={\aleph }_{\alpha +1}$. Here $2^{\kappa }$ is the cardinality of the power set of a set of cardinality $\kappa$, and ${\kappa }^{+}$ is the least strictly larger cardinal. The hypothesis says that no intermediate cardinal lies between an infinite set and its power set.