= Cofinality of a power-set cardinal
{title2=$\operatorname{cf}(2^\kappa)>\kappa$}
For an infinite <cardinal number> $\kappa$, use <cardinal arithmetic> with the <axiom of choice>. If $\lambda=2^\kappa$ had <cofinality> $\mu\leq\kappa$, choose cofinal <ordinals> $\alpha_i<\lambda$, $i<\mu$. Their <cardinalities> $a_i$ satisfy $\sum_i a_i=\lambda$. The <König theorem for cardinal numbers> gives $\lambda<\prod_i a_i^+\leq\lambda^\mu\leq(2^\kappa)^\kappa=\lambda$, a contradiction.
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