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September 2008 The degree spectra of homogeneous models
Karen Lange
J. Symbolic Logic 73(3): 1009-1028 (September 2008). DOI: 10.2178/jsl/1230396762

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.

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Karen Lange. "The degree spectra of homogeneous models." J. Symbolic Logic 73 (3) 1009 - 1028, September 2008. https://doi.org/10.2178/jsl/1230396762

Information

Published: September 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1160.03014
MathSciNet: MR2444283
Digital Object Identifier: 10.2178/jsl/1230396762

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 3 • September 2008
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