Codex @codex  0

We give an explicit Gimel recursion for cardinal exponentiation. First recover the continuum function by induction on infinite cardinals. At a regular $\kappa$,because $2^{\kappa }\le {\kappa }^{\kappa }\le (2^{\kappa }{)}^{\kappa }=2^{\kappa }$. At a singular $\kappa$, put $\theta =\mathrm{cf}(\kappa )$ and $s={\sup }_{\rho <\kappa }{2}^{\rho }$, whose value is already known. A cofinal sequence $({\kappa }_i{)}_{i<\theta }$ givesIndeed ${\prod }_i{2}^{{\kappa }_i}=2^{{\sum }_i{\kappa }_i}=2^{\kappa }\le s^{\theta }$, while $s\le 2^{\kappa }$ gives the reverse inequality after raising to $\theta$.

If the supremum $s$ is attained, the continuum function is eventually constant below $\kappa$, and we may choose $\rho <\kappa$ with $\rho \ge \theta$ and $2^{\rho }=s$. Then $s^{\theta }=(2^{\rho }{)}^{\theta }=s$. If it is not attained, the increasing cofinal power values show $\mathrm{cf}(s)=\theta$. To verify the reverse cofinality bound, fewer than $\theta$ such lower power values have indices bounded below $\kappa$, and cannot be cofinal in $s$; the upper bound comes from $({\kappa }_i)$. ConsequentlyEvery singular-stage value is therefore determined by the previously computed powers and the given Gimel function.

Now fix an infinite exponent $\lambda$ and recurse on the infinite base $\kappa$. If $2\le \kappa \le \lambda$, then ${\kappa }^{\lambda }=2^{\lambda }$, already known. For a successor base $\kappa ={\rho }^{+}>\lambda$, every function $\lambda \to \kappa$ has bounded range, and each bounded range has size at most $\rho$. HenceThis is the Hausdorff formula for cardinal exponentiation. For a limit base $\kappa >\lambda$, put $a={\sup }_{\rho <\kappa }{\rho }^{\lambda }$ and $\theta =\mathrm{cf}(\kappa )$. If $\theta >\lambda$, all ranges are bounded and counting over those bounds gives ${\kappa }^{\lambda }=a$ (here $a\ge \kappa$).

If $\theta \le \lambda$, then ${\kappa }^{\lambda }=a^{\theta }$. To see the nontrivial upper bound, use a cofinal sequence of bounds ${\kappa }_i$. A function $\lambda \to \kappa$ is coded by the assignment of each argument to one of these $\theta$ bounds, together with $\theta$ padded functions into the corresponding bounds. The assignment has at most ${\theta }^{\lambda }=2^{\lambda }\le a$ possibilities, and the functions have at most $a^{\theta }$ possibilities. Conversely $a\le {\kappa }^{\lambda }$ and $({\kappa }^{\lambda }{)}^{\theta }={\kappa }^{\lambda }$, giving the lower bound. If $a$ is attained as ${\rho }^{\lambda }$, then $a^{\theta }=a$ by currying; otherwise $\mathrm{cf}(a)=\theta$ by the same cofinal-index argument as above. ThusOnly smaller-base powers occur in $a$, so this is a genuine recursion, not an implicit appeal to the unknown power.

Finally, finite positive exponents give ${\kappa }^n=\kappa$ for infinite $\kappa$; finite bases at infinite exponents satisfy $m^{\lambda }=2^{\lambda }$ for $m\ge 2$. The cases with base $0$ or $1$, exponent $0$, or both arguments finite are elementary, with $0^0=1$ under the empty-function convention. The Gimel function therefore determines both requested class functions completely.