A note on direct sums of Friedbergnumberings

Extract

We show that a translator ƒ: ω → ω from a Gödelnumbering φ into a direct sum η of a r.e. family of Friedbergnumberings satisfies ƒ ≰_T0′. In particular, η cannot be a Gödelnumbering.

In the following we use standard notation (cf.[3]): for i ≥ 1, P_i (respectively, R_i) is the set of partial (total) recursive i-place functions; φ is a Gödel numbering of P₁. By φ_{i, s} we denote a recursive standard approximation for φ_i, i.e., φ_{i, s} is a finite function, φ_{i, s} ⊆ φ_{i, s + 1}, φ_{i, s} ⊆ φ_i, φ_i = ⋃ {φ_{i, s} ∣ s ≥ 0}, and a canonical index for φ_{i, s} can be computed uniformly in i, s (cf.[3, p. 16 f]).

We call v ∈ P₂ a numbering of P₁ iff {λx.v(i, x)}_{i ∈ ω} = P₁; we denote λx.v(i, x) by v_i. Let v and γ be numberings of P₁. A total function g: ω → ω is called a translator from v into γ iff ∀i: v_i = γ_g(i).