Biinterpretability up to double jump in the degrees below 0′

Shore, Richard

doi:10.1090/S0002-9939-2013-11719-3

Biinterpretability up to double jump in the degrees below $\mathbf {0}^{\prime }$
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by Richard A. Shore

Proc. Amer. Math. Soc. 142 (2014), 351-360

DOI: https://doi.org/10.1090/S0002-9939-2013-11719-3

Published electronically: September 12, 2013

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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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