Genericity and Large Cardinals

A natural question to ask is whether this result has an analogue in the context of large cardinals. The purpose of this article is to provide the strongest such analogue not ruled out by limitations imposed by the existence of Woodin cardinals. To describe the latter limitations we consider the forcing P , described as follows. Let δ be inaccessible and consider the language L(δ): (a) n ∈ R belongs to L(δ), where n ∈ ω and R denotes a real. (b) φ ∈ L(δ) →∼ φ ∈ L(δ). (c) Φ ⊆ L(δ), card Φ < δ → ∧Φ ∈ L(δ). Of course ∧Φ is to be interpreted as the conjunction of the sentences in Φ. A set of sentences Φ ⊆ L(δ) is consistent if in some set-generic extension, some real R satisfies each sentence in Φ. A single sentence φ ∈ L(δ) is consistent if {φ} is consistent. We endow L(δ) with the ordering φ ≤ ψ iff ∧{φ,∼ ψ} is not consistent. Then P is the pre-ordering (L(δ),≤) where L(δ) = {φ ∈ L(δ) | φ is consistent}.