Geodesic
DNR AND INCOMPARABLE TURING DEGREES
Forum of Mathematics, Sigma, Tome 4 (2016)

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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. 
@article{10_1017_fms_2016_3,
     author = {MINZHONG CAI and NOAM GREENBERG and MICHAEL MCINERNEY},
     title = {DNR {AND} {INCOMPARABLE} {TURING} {DEGREES}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2016},
     doi = {10.1017/fms.2016.3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2016.3/}
}
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MINZHONG CAI; NOAM GREENBERG; MICHAEL MCINERNEY. DNR AND INCOMPARABLE TURING DEGREES. Forum of Mathematics, Sigma, Tome 4 (2016). doi: 10.1017/fms.2016.3

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