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

Abstract

We prove that, for every z 00 with z00 > 000 (i.e. z 2 L2), the structure D( z) of the Turing degrees below x is biinterpretable with rst rst order arithmetic up to double jump. As a corollary, every relation on D( z) which is invariant under double jump is de nable in D( z) if and only if it is de nable in arithmetic.