The degree spectra of homogeneous models

Abstract

Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model 𝒜 has a d-basis if the types realized in 𝒜 are all computable and the Turing degree d can list Δ₀⁰-indices for all types realized in 𝒜. 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 𝒜 with a 0-basis but no decidable copy. We prove that any homogeneous 𝒜 with a 0’-basis has a low decidable copy. This implies Csima’s analogous result for prime models. In the case where all types of the theory T are computable and 𝒜 is a homogeneous model with a 0-basis, we show 𝒜 has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous 𝒜 with a 0-basis has a d-decidable copy. We show that the nonlow₂ Δ₂⁰ degrees are 0-homogeneous bounding.