DNR AND INCOMPARABLE TURING DEGREES

Abstract

We construct an increasing ${\it\omega}$-sequence $\langle \boldsymbol{a}_{n}\rangle$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each $\boldsymbol{a}_{n+1}$ is diagonally nonrecursive relative to $\boldsymbol{a}_{n}$. It follows that the DNR principle of reverse mathematics does not imply the existence of Turing incomparable degrees.