Cardinal exponentiation (${\kappa }^{\lambda }$)

Cardinal exponentiation counts functions from a set of size $\lambda$ to one of size $\kappa$. The currying law for cardinal exponentiation gives $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \cdot \mu }$. These are cardinal operations, distinct from ordinal exponentiation. For example the Generalized continuum hypothesis specifies $2^{\kappa }={\kappa }^{+}$ at infinite cardinals.

Assume the Generalized continuum hypothesis. For necessity, König theorem for cardinal numbers gives ${\rho }^{\mathrm{cf}(\rho )}>\rho$ for every infinite cardinal. But $\lambda ={\kappa }^{\mu }$ implies ${\lambda }^{\mu }=\lambda$ by the currying law for cardinal exponentiation, so $\mathrm{cf}(\lambda )>\mu$. 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 ${\nu }^{{\aleph }_0}\le 2^{\nu }={\nu }^{+}\le \lambda$. There are only $\lambda$ ordinal bounds to consider, giving ${\lambda }^{{\aleph }_0}=\lambda$ and the representation with base $\lambda$ and exponent ${\aleph }_0$.