= Solution
In the usual arithmetic under the <axiom of choice> of <cardinal numbers>, take the singular strong-limit <cardinal>
$$
\kappa=\beth_\omega=\sup_{n<\omega}\beth_n,\qquad
\beth_0=\aleph_0,\quad\beth_{n+1}=2^{\beth_n}.
$$
<Cantor's theorem> makes this sequence strictly increasing. It is <countable> and <cofinal> in its <supremum>, so $\operatorname{cf}(\kappa)\le\aleph_0$. No finite <subset> is <cofinal> in this <limit ordinal>, hence $\operatorname{cf}(\kappa)=\aleph_0$. Therefore
$$
\boxed{2^{\operatorname{cf}(\kappa)}=2^{\aleph_0}=\beth_1<\beth_\omega=\kappa.}
$$
This 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.
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