A Friedberg-Muchnik Type Theorem from Recursive Model Theory

In [1], Knight gave conditions on recursive L-structure A with a pair of recursive subsets R and S, under which there is an isomorphism f from A to B such that f (R) and f (S) are Σ0/β and independent over ∆0/β, where β ≥ 1 is a recursive ordinal. In response to a question of Knight which was motivated by [1], we give similar conditions on a recursive L-structure A with two pairs of resursive subsets Ri+1, Si-1 (i = 0, 1), which which there exists an isomorphism f from A onto a recursive L-structure B, such that f (Ri+1), f (Si+1) are Σ0/i+1 sets with Turing degrees incomparable relative to 0(i).

We treat this 0″-priority argument as a test case for a level-two metatherorem--a one-step generalization of a metatheorem for finite injury priority arguments given in [2].