The degree spectra of homogeneous models

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 0 0-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 homogenous A with a 0-basis has a d-decidable copy. W...

• Type: Journal
• Year: 2008
• Where: JSYML
• Authors: Karen Lange
• Keywords: Decidable Copy, Homogeneous Model, JSYML 2006, JSYML 2008, Prime Models