We isolate a forcing which increases the value of δ12 while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.

We isolate a forcing which increases the value of $\delta _{2}^{1}$ while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.

We show that a Woodin cardinal is necessary for the extender algebra to be complete. Our proof is relatively simple and does not use fine structure.

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ 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 $\omega _{1}^{V}$ such that M is countably closed in the forcing extension. (shrink)