A characterization of the 0-basis homogeneous bounding degrees

Abstract

We say a countable model 𝒜 has a 0-basis if the types realized in 𝒜 are uniformly computable. We say 𝒜 has a (d-)decidable copy if there exists a model ℬ≅𝒜 such that the elementary diagram of ℬ is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model 𝒜 with a 0-basis but no decidable copy. We extend this result here. Let d≤0' be any low₂ degree. We show that there exists a homogeneous model 𝒜 with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous 𝒜 with a 0-basis has a d-decidable copy. In previous work, we showed that the nonlow₂ Δ₂⁰ degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding Δ₂⁰ degrees.