Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 5 / c / Solution

The aleph numbers enumerate the infinite initial ordinals by transfinite recursion:Here ${\kappa }^{+}$ is the least cardinal strictly larger than $\kappa$. The Generalized continuum hypothesis isequivalently $2^{{\aleph }_{\alpha }}={\aleph }_{\alpha +1}$ for every ordinal $\alpha$ in ZFC.