Defining jump classes in the degrees below $\textbf {0}’$

Abstract:

We prove that, for each degree ${\mathbf {c}}$ r.e. in and above ${{\mathbf {0}}^{(3)}}$, the class of degrees ${\mathbf {x}} \leq {\mathbf {0}}’$ with ${{\mathbf {x}}^{(3)}} = {\mathbf {c}}$ is definable without parameters in $\mathcal {D}( \leq 0’)$, the degrees below ${\mathbf {0’}}$. Indeed the same definitions work below any r.e. degree ${\mathbf {r}}$ in place of ${\mathbf {0’}}$. Thus for each r.e. degree ${\mathbf {r}}$, $\operatorname {Th} (\mathcal {D}( \leq {\mathbf {r}}))$ uniquely determines ${{\mathbf {r}}^{(3)}}$.