Computability: Recovering Delta_0 from Sigma_1?

From: Rex Butler (RexButler_at_hotmail.com)
Date: 11/17/04 

Date: 16 Nov 2004 19:46:17 -0800

This question is necessarily vague, but is there a known way to
'recover' the set of all -primitive- recursive relations (Delta_0)
from the set of recursively enumerable relations (Sigma_1)? Going up
the arithmetical hierarchy is easy, going down not so.

Note: I use the convention that Delta_0 is the set of primitive
recursive relations, not the set of computable relations. In
particular Delta_0 is not equal to Delta_1. Either way, the resulting
hierarchies are the same everywhere except of course at Delta_0. I
posted a thread asking about this choice a while back, see "Primitive
recursive vs recursive at the base of the arithmetical hierarchy."

Rex Butler