We isolate a forcing which increases the value of δ 2 while preserving ω1 under the assumption that there is a precipitous ideal on ω1 and a measurable cardinal.

We present Woodin’s proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all Σ2 sentences φ such that CH+φ holds in a forcing extension of V by a… (More)

Assume AD and that either V = L(P(R)), or V = L(T, R) for some set T ⊂ ORD. Let (X,≤) be a pre-partially ordered set. Then exactly one of the following cases holds: (1) X can be written as a… (More)

Woodin and Steel showed that under AD + DCR the Suslin cardinals are closed below their supremum; Woodin devised an argument based on the notion of strong ∞-Borel code which is presented here. A… (More)

Using ♦ and large cardinals we extend results of Magidor–Malitz and Farah–Larson to obtain models correct for the existence of uncountable homogeneous sets for finite-dimensional partitions and… (More)

Theorem 0.2. Suppose that there is a stationary S ⊂ ω1 such that NSω11S is saturated. Then there is a forcing preserving stationary subsets of S which adds a collection {Cn : n ∈ ω} of club subsets… (More)