Journal of Symbolic Logic

The degree spectra of homogeneous models

Karen Lange

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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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J. Symbolic Logic, Volume 73, Issue 3 (2008), 1009-1028.

First available in Project Euclid: 27 December 2008

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Lange, Karen. The degree spectra of homogeneous models. J. Symbolic Logic 73 (2008), no. 3, 1009--1028. doi:10.2178/jsl/1230396762.

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