Power set at ${\mathrm{\aleph }}_{\omega }$: On a theorem of Woodin

We give Woodin’s original proof that if there exists a $(\kappa +2)-$strong cardinal $\kappa ,$ then there is a generic extension of the universe in which $\kappa ={\mathrm{\aleph }}_{\omega },$ $GCH$ holds below ${\mathrm{\aleph }}_{\omega }$ and $2^{{\mathrm{\aleph }}_{\omega }}={\mathrm{\aleph }}_{\omega +2}\mathrm{.}$