Limits to joining with generics and randoms

TL;DR

This paper investigates the limitations of constructing generic and random sets in computability theory, showing that certain sets cannot be weakly 2-generic or 2-random, highlighting fundamental constraints in effective forcing methods.

Contribution

It proves that the sets obtained in Shore-Slaman theorems cannot be weakly 2-generic or 2-random, revealing inherent limitations in the use of effective forcing in computability.

Findings

Sets in Shore-Slaman theorems are not weakly 2-generic.

Such sets cannot be 2-random.

Results apply to various effective forcing notions.

Abstract

Posner and Robinson (1981) proved that if $S \subseteq \omega$ is non-computable, then there exists a $G \subseteq \omega$ such that $S \oplus G \geq_T G'$. Shore and Slaman (1999) extended this result to all $n \in \omega$, by showing that if $S \nleq_T \emptyset^{(n-1)}$ then there exists a $G$ such that $S \oplus G \geq_T G^{(n)}$. Their argument employs Kumabe-Slaman 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 $n \geq 1$, 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.