= Gimel recursion for cardinal exponentiation
{title2=$\gimel\Longrightarrow(\kappa,\lambda)\mapsto\kappa^\lambda$}
The <Gimel function> determines all infinite <cardinal> powers. For a regular base, $2^\kappa=\gimel(\kappa)$. For a singular base, put $s=\sup_{\rho<\kappa}2^\rho$ and $\theta=\operatorname{cf}(\kappa)$; then $2^\kappa=s^\theta$, which is $s$ if attained below $\kappa$ and $\gimel(s)$ otherwise. After these powers are known, fix an infinite exponent $\lambda$ and recurse on the base. Below $\lambda$ use $2^\lambda$; at successors use the <Hausdorff formula for cardinal exponentiation>. At a limit base greater than $\lambda$, put $a=\sup_{\rho<\kappa}\rho^\lambda$. The answer is $a$ if $\operatorname{cf}(\kappa)>\lambda$ or the supremum is attained, and $\gimel(a)$ otherwise. In the nonattained cases, the cofinal-index argument identifies the <cofinality> of the supremum.
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