Pseudojump inversion in special r. b. ${\Pi }_1^0$ classes

Abstract

The Jump Inversion Theorem says that for every real $A{\ge }_T{0}^{\prime }$ there is a real $B$ such that $A{\equiv }_T{B}^{\prime }{\equiv }_{T}B\oplus 0^{\prime }$. A known refinement of this theorem says that we can choose $B$ to be a member of any special ${\Pi }_1^0$ subclass of $\{0,1{\}}^{\omega }$. We now consider the possibility of analogous refinements of two other well-known theorems: the Join Theorem -- for all reals $A$ and $Z$ such that $A{\ge }_{T}Z\oplus 0^{\prime }$ and $Z{>}_{T}0$, there is a real $B$ such that $A{\equiv }_T{B}^{\prime }{\equiv }_{T}B\oplus 0^{\prime }{\equiv }_{T}B\oplus Z$ -- and the Pseudojump Inversion Theorem -- for all reals $A{\ge }_T{0}^{\prime }$ and every $e\in \omega$, there is a real $B$ such that $A{\equiv }_{T}B\oplus W_e^B{\equiv }_{T}B\oplus 0^{\prime }$. We show that in these theorems, $B$ can be found in some special ${\Pi }_1^0$ subclasses of $\{0,1{\}}^{\omega }$ but not in others.

19 pages, submitted for publication