I'm going to assume GCH here. If that holds, then why do we have the equalities $2^{\lt\kappa}=\kappa$ for every $\kappa$, and $\kappa^{\lt\kappa}=\kappa$ for all regular $\kappa$?
Why do $2^{\lt\kappa}=\kappa$ and $\kappa^{\lt\kappa}=\kappa$ when the Generalized Continuum Hypothesis holds?
1 Answers
Recall the definition of $\lambda^{<\mu} = \sup\{\lambda^\nu\mid\nu<\mu\}$.
Assume GCH, let $\kappa$ be a regular cardinal.
$2^{<\kappa} = \sup\{2^\lambda \mid \lambda<\kappa\} = \sup\{\lambda^+\mid\lambda<\kappa\} = \kappa$
Where the last equality holds since if $\kappa$ is a successor cardinal then it is $\lambda^+$ for some $\lambda<\kappa$; and if it is a regular limit cardinal then it is the limit of successor cardinals below it.
$\kappa^{<\kappa} = \sup\{\kappa^\lambda\mid\lambda<\kappa\} = \kappa$
The last equality follows from: $\begin{align} \kappa^\lambda &=\kappa\cdot\sup\{\mu^\lambda\mid\mu<\kappa\}\\ &\le\kappa\cdot\sup\{\max\{2^\mu,2^\lambda\}\mid\mu<\kappa\}\\ &=\kappa\cdot\sup\{\max\{\mu^+,\lambda^+\}\mid\mu<\kappa\}\\ &=\kappa\cdot\kappa=\kappa \end{align}$
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1Yeah, you're right. I have no idea why I wrote that in the definition. Oh well. – 2014-05-08