cardinal exponentiation under GCH

Many results about cardinal exponentiation can neither be proved nor disproved in ZFC. If, however, we allow ourselves to use GCH in addition to ZFC, then we have the following theorem, which gives an essentially complete description of the way cardinal exponentiation involving infinite cardinals works.

Theorem.

Assume the Generalized Continuum Hypothesis holds. Let $\kappa$ and $\lambda$ be cardinals, at least one of which is infinite, and such that $\kappa>0$ and $\lambda>1$ . Then

$\lambda^{\kappa}=\begin{cases}\kappa^{+},&\text{if}\quad\lambda\leq\kappa^{+};% \\ \lambda^{+},&\text{if}\quad\operatorname{cf}(\lambda)\leq\kappa\leq\lambda;\\ \lambda,&\text{if}\quad\kappa<\operatorname{cf}(\lambda).\end{cases}$

Here, $\operatorname{cf}(\lambda)$ is the cofinality of $\lambda$ , and $\lambda^{+}$ is the cardinal successor of $\lambda$ .

Title	cardinal exponentiation under GCH
Canonical name	CardinalExponentiationUnderGCH
Date of creation	2013-03-22 14:54:04
Last modified on	2013-03-22 14:54:04
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Theorem
Classification	msc 03E10
Related topic	CardinalArithmetic
Related topic	GeneralizedContinuumHypothesis