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The degree spectra of homogeneous models
Published online by Cambridge University Press: 12 March 2014
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 has a d-basis if the types realized in
are all computable and the Turing degree d can list
-indices for all types realized in
. We say
has a d-decidable copy if there exists a model
≅
such that the elementary diagram of
is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous
with a 0-basis but no decidable copy.
We prove that any homogeneous 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
is a homogeneous model with a 0-basis, we show
has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous
with a 0-basis has a d-decidable copy. We show that the nonlow2
degrees are 0-homogeneous bounding.
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- Copyright © Association for Symbolic Logic 2008
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