Biinterpretability up to double jump in the degrees below $\mathbf {0}^{\prime }$
Richard A. Shore · Proceedings of the American Mathematical Society · 2013
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.