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, Pi (respectively, Ri) is the set of partial (total) recursive i-place functions; φ is a Gödel numbering of P1. By φi, s we denote a recursive standard approximation for φi, i.e., φi, s is a finite function, φi, s ⊆ φi, s + 1, φi, s ⊆ φi, φi = ⋃ {φi, s ∣ s ≥ 0}, and a canonical index for φi, 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 vi. Let v and γ be numberings of P1. A total function g: ω → ω is called a translator from v into γ iff ∀i: vi = γg(i).