Under what assumptions on an infinite cardinal $\kappa$ we have $\kappa^\kappa= 2^\kappa?$
Please delete this question. I know the answer.
Under what assumptions on an infinite cardinal $\kappa$ we have $\kappa^\kappa= 2^\kappa?$
Please delete this question. I know the answer.
Assuming $\kappa$ is an ordinal the answer is always.
The reason is simple: by Cantor's theorem we have 2<\kappa<2^\kappa, therefore using exponentiation laws: $2^\kappa\le\kappa^\kappa\le\left(2^\kappa\right)^\kappa=2^{\kappa\cdot\kappa}=2^\kappa$