Omitting cohesive sets

TL;DR

The paper demonstrates limitations on the computational power of certain generic and random sets in computing cohesive sets for specific computable sequences, highlighting inherent computational boundaries.

Contribution

It establishes that no 3-generic or Martin-Löf random set can compute cohesive sets for sequences lacking computable cohesive sets.

Findings

No 3-generic computes any $oldsymbol{R}$-cohesive set.

There exists a Martin-Löf random that computes no $oldsymbol{R}$-cohesive set.

Results reveal limitations of generic and random sets in computing certain cohesive sets.

Abstract

We prove that if $\vec{R}$ is a computable sequence of subsets of $\omega$ which admits no computable cohesive set, then no 3-generic computes any $\vec{R}$-cohesive set; and there exists a Martin-L\"{o}f random which computes no $\vec{R}$-cohesive set.