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

For GCH arithmetic at aleph omega, let $\kappa ={\aleph }_{\omega }$. Use monotonicity of cardinal exponentiation, the law $(a^b{)}^c=a^{bc}$, and $\kappa \cdot \kappa =\kappa$ for infinite cardinals in ZFC. Since $2\le \kappa \le 2^{\kappa }$,Thus ${\kappa }^{\kappa }=2^{\kappa }$, and the generalized continuum hypothesis yieldsThe lower bound embeds binary-valued functions into $\kappa$-valued ones; the upper bound codes each value by a subset and combines the resulting pairs of indices. In particular the equality before applying the hypothesis is a ZFC equality.