Limits to joining with generics and randoms

Abstract

Abstract. 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 2-generic. 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.

Decomposing Borel functions using the Shore-Slaman join theorem

“…Historically, Jockusch and Shore [16] were the first to ask whether the Posner-Robinson join theorem can be generalized to all n-REA operators for n ∈ ω. The main tool for addressing their question was introduced by Kumabe and Slaman, who showed the join theorem for α = ω (for Kumabe-Slaman forcing, see also Day-Dzhafarov [5]). Finally, Shore and Slaman proved the join theorem for all computable ordinals α.…”

Higher Randomness and Genericity

“…Their solution gives yet another characterization of lowness for ML-randomness. Limits on cupping with random sequences were established by Day and Dzhafarov [DD13].…”