DNR and incomparable Turing degrees

TL;DR

This paper constructs a specific sequence of Turing degrees demonstrating that the DNR principle in reverse mathematics does not guarantee the existence of Turing incomparable degrees, revealing limitations in the logical implications of DNR.

Contribution

It introduces a novel construction of an increasing sequence of Turing degrees with specific noncomputability properties, showing a new separation in reverse mathematics.

Findings

Constructs an increasing sequence of Turing degrees with specific properties.

Shows DNR does not imply the existence of Turing incomparable degrees.

Provides a counterexample in reverse mathematics.

Abstract

We construct an increasing $\omega$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\mathsf{DNR}$ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.