= Infinite cardinal powers under GCH
{c}
{title2=$\lambda=\kappa^\mu\text{ for infinite }\kappa,\mu\iff\operatorname{cf}(\lambda)>\aleph_0$}
Assume the <Generalized continuum hypothesis>. For necessity, <König theorem for cardinal numbers> gives $\rho^{\operatorname{cf}(\rho)}>\rho$ for every infinite <cardinal>. But $\lambda=\kappa^\mu$ implies $\lambda^\mu=\lambda$ by the <currying law for cardinal exponentiation>, so $\operatorname{cf}(\lambda)>\mu$. This direction uses only <ZFC>. Conversely, uncountable <cofinality> bounds the range of every <countable> sequence in $\lambda$. Under <GCH>, each smaller infinite <cardinal> $\nu$ satisfies $\nu^{\aleph_0}\leq2^\nu=\nu^+\leq\lambda$. There are only $\lambda$ <ordinal> bounds to consider, giving $\lambda^{\aleph_0}=\lambda$ and the representation with base $\lambda$ and exponent $\aleph_0$.
Back to article page
