A Bounded Jump for the Bounded Turing Degrees

Abstract

We define the bounded jump of $A$ by $A^b=\{x\in \omega \mid \exists i\le x[{\phi }_i\left(x\right)\downarrow \wedge {\Phi }_x^{A\upharpoonright \upharpoonright {\phi }_i\left(x\right)}\left(x\right)\downarrow \left]\right\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact that it is strictly increasing and order-preserving on the bounded Turing ($bT$) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for $n\ge 2$ we have $X{\le }_{bT}{\varnothing }^{nb}\iff X$ is ${\omega }^n$-c.e. $\iff X{\le }_1{\varnothing }^{nb}$, extending the classical result that $X{\le }_{bT}\varnothing \text{'}\iff X$ is $\omega$-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every $X$ such that ${\varnothing }^b{\le }_{bT}X{\le }_{bT}{\varnothing }^{2b}$, there is a $Y{\le }_{bT}{\varnothing }^b$ such that $Y^b{\equiv }_{bT}X$.