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 and be cardinals,at least one of which is infinite,and such that and .Then
Here, is the cofinality of , and is the cardinal successor of .