Regular embeddings of the stationary tower and Woodin's Σ22 maximality theorem

Abstract

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all Σ²₂ sentences φ such that CH + φ holds in a forcing extension of V by a partial order in V_δ. We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding j : V → M with critical point ω₁^V such that M is countably closed in the forcing extension.