A Simple Proof of Silver's Theorem

Abstract

By using combinatorial properties of stationary sets, we give a
simple proof of some generalization of Silver's Theorem i.e. if
$\kappa$ is an uncountable regular cardinal such that
$\aleph_\kappa$ is a singular strong limit cardinal, then the
following hold.

(1). If $\{\alpha<\kappa : \aleph_\alpha^{<\kappa} \leq
\aleph_{\alpha\cdot2}\}$ is stationary, then $2^{\aleph_{\kappa}}
\leq \aleph_{\kappa\cdot2}$.

(2). If $\{\alpha<\kappa : \aleph_\alpha^{<\kappa} \leq
\aleph_{\alpha+\gamma}\}$, where $0<\gamma<\kappa$,  is stationary,
then $2^{\aleph_{\kappa}} \leq \aleph_{\kappa+\gamma}$.