 Solution

In the usual arithmetic under the axiom of choice of cardinal numbers, take the singular strong-limit cardinalCantor's theorem makes this sequence strictly increasing. It is countable and cofinal in its supremum, so $\mathrm{cf}(\kappa )\le {\aleph }_0$. No finite subset is cofinal in this limit ordinal, hence $\mathrm{cf}(\kappa )={\aleph }_0$. ThereforeThis construction needs no assumption such as the Continuum hypothesis. The conventional use of cardinals here includes well-orderability of these power sets, supplied by the axiom of choice. It is a separate setting from the explicit failure of well-orderability hypothesized in question 8; one must not impose choice on that later hypothesis.