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Biinterpretability up to double jump in the degrees below $ \mathbf{0}^{\prime}$

Author: Richard A. Shore
Journal: Proc. Amer. Math. Soc. 142 (2014), 351-360
MSC (2010): Primary 03D28
Published electronically: September 12, 2013
MathSciNet review: 3119208
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Abstract: We prove that for every $ \mathbf {z\leq 0}^{\prime }$ with $ \mathbf {z}^{\prime \prime }>\mathbf {0}^{\prime \prime }$ (i.e. $ \mathbf {z\in \bar {L}}_{2}$), the structure $ \mathcal {D}(\leq \mathbf {z})$ of the Turing degrees below $ \mathbf {x}$ is biinterpretable with first order arithmetic up to double jump. As a corollary, every relation on $ \mathcal {D}(\leq \mathbf {z})$ which is invariant under double jump is definable in $ \mathcal {D}(\leq \mathbf {z})$ if and only if it is definable in arithmetic.

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Richard A. Shore
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Received by editor(s): December 27, 2011
Received by editor(s) in revised form: December 28, 2011, and February 26, 2012
Published electronically: September 12, 2013
Additional Notes: The author was partially supported by NSF Grants DMS-0852811 and DMS-11675, and as a Visiting Professor in the Department of Mathematics and the Institute for Mathematical Sciences at the National University of Singapore, with partial funding from the John Templeton Foundation.
Communicated by: Julia Knight
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.