Limits to Joining with Generics and Randoms

Posner and Robinson [4] proved that if S ⊆ ω is non-computable, then there exists a G ⊆ ω such that S ⊕ G ≥T G′. Shore and Slaman [7] extended this result to all n ∈ ω, by showing that if S T ∅(n−1) then there exists a G such that S ⊕ G ≥T G(n). Their argument employs KumabeSlaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. We answer the question of whether this is a necessary complication by showing that for all n ≥ 1, the set G of the Shore-Slaman theorem cannot be chosen to be even weakly 2generic. Our result applies to several other effective forcing notions commonly used in computability theory, and we also prove that the set G cannot be chosen to be 2-random.