Recursion Theory

5.3.2 Sacks generic reals

A set D ⊆ is dense if for any condition T, there is an S ∈ D such that S ≤ T. A real x is Sacks generic if for any dense set D ⊆ arithmetical in , there is a condition T ∈ such that x ∈ [T]. Note that by Lemma 5.3.2 and Lemma 5.3.4, for any sentence φ the set Dφ = {T | T ⊩ φ ∨ T ⊩¬φ} is dense and arithmetical in .

Lemma ...