Splitting an $\alpha$-recursively enumerable set

Abstract:

We extend the priority method in $\alpha$-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible $\alpha$. THEOREM: Let $C$ be a regular $\alpha$-r.e. set and $D$ be a nonrecursive $\alpha$-r.e. set. Then there are regular $\alpha$-r.e. sets $A$ and $B$ such that $A \cup B = C,A \cap B = \phi ,A,B{ \leq _\alpha }C$ and such that $D$ is not $\alpha$-recursive in $A$ or $B$. The result is also strengthened to apply to ${ \leq _{c\alpha }}$, and various corollaries about the structure of the $\alpha$ and $c\alpha$ recursively enumerable degrees are proved.