Gimel function ($\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$)

For an infinite cardinal number, $\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$. The singular cardinals hypothesis predicts its value when $\kappa$ is a singular cardinal and $2^{\mathrm{cf}(\kappa )}<\kappa$. For a regular cardinal, its value is $2^{\kappa }$.

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 $\theta =\mathrm{cf}(\kappa )$; 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 $\mathrm{cf}(\kappa )>\lambda$ or the supremum is attained, and $\gimel (a)$ otherwise. In the nonattained cases, the cofinal-index argument identifies the cofinality of the supremum.

For every singular infinite cardinal number $\kappa$, the Gimel function has the smallest value permitted by König theorem for cardinal numbers and the exponent: ${\kappa }^{\mathrm{cf}(\kappa )}=\max \{{\kappa }^{+},2^{\mathrm{cf}(\kappa )}\}$. Where $2^{\mathrm{cf}(\kappa )}<\kappa$, this is the singular cardinals hypothesis. The hypothesis imposes no separate successor-power condition on regular cardinals.