Power set at $\aleph_\omega$: On a theorem of Woodin

TL;DR

This paper presents a detailed proof demonstrating that the existence of a specific strong cardinal implies the possibility of a universe extension where the continuum at  is precisely +2, with GCH holding below .

Contribution

It provides Woodin's original proof linking strong cardinals to the structure of the continuum at .

Findings

Existence of a (+2)-strong cardinal implies a universe extension with =.

In this extension, GCH holds below  and the continuum at  is +2.

The proof confirms the consistency of specific continuum configurations under large cardinal assumptions.

Abstract

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 $\kappa=\aleph_\omega,$ $GCH$ holds below $\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}.$