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DNR AND INCOMPARABLE TURING DEGREES

  • MINZHONG CAI (a1), NOAM GREENBERG (a2) and MICHAEL MCINERNEY (a2)
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.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
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[1] Ambos-Spies, K., Kjos-Hanssen, B., Lempp, S. and Slaman, T. A., ‘Comparing DNR and WWKL’, J. Symbolic Logic 69(4) (2004), 10891104.
[2] Cai, M., ‘A 2-minimal non-GL2 degree’, J. Math. Log. 10(1–2) (2010), 130.
[3] Cai, M., ‘A hyperimmune minimal degree and an ANR 2-minimal degree’, Notre Dame J. Form. Log. 51(4) (2010), 443455.
[4] Cai, M., ‘Elements of classical recursion theory: degree-theoretic properties and combinatorial properties’, PhD thesis, Cornell University, 2011.
[5] Cai, M., ‘2-minimality, jump classes and a note on natural definability’, Ann. Pure Appl. Logic 165(2) (2014), 724741.
[6] Conidis, C. J., ‘A measure-theoretic proof of Turing incomparability’, Ann. Pure Appl. Logic 162(1) (2010), 8388.
[7] Greenberg, N. and Miller, J. S., ‘Diagonally non-recursive functions and effective Hausdorff dimension’, Bull. Lond. Math. Soc. 43(4) (2011), 636654.
[8] Jockusch, C. G. Jr, ‘Degrees of functions with no fixed points’, inLogic, Methodology and Philosophy of Science, VIII (Moscow, 1987), Studies in Logic and the Foundations of Mathematics, 126 (North-Holland, Amsterdam, 1989), 191201.
[9] Khan, M. and Miller, J. S., Forcing with bushy trees (in preparation).
[10] Kučera, A. and Slaman, T. A., ‘Turing incomparability in Scott sets’, Proc. Amer. Math. Soc. 135(11) (2007), 37233731.
[11] Kumabe, M. and Lewis, A. E. M., ‘A fixed-point-free minimal degree’, J. Lond. Math. Soc. (2) 80(3) (2009), 785797.
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Forum of Mathematics, Sigma
  • ISSN: -
  • EISSN: 2050-5094
  • URL: /core/journals/forum-of-mathematics-sigma
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