DNR and Incomparable Turing Degrees
Forum of Mathematics Sigma
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 nonrecursive relative to a(n). It follows that the DNR principle of reverse mathematics does not imply the existence of Turing incomparable degrees.
Cai, Minzhong; Greenberg, Noam; and Mcinerney, Michael, "DNR and Incomparable Turing Degrees" (2016). Open Dartmouth: Faculty Open Access Articles. 395.