Journal of Symbolic Logic

A characterization of the 0-basis homogeneous bounding degrees

Karen Lange

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We say a countable model 𝒜 has a 0-basis if the types realized in 𝒜 are uniformly computable. 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 model 𝒜 with a 0-basis but no decidable copy. We extend this result here. Let d0' be any low₂ degree. We show that there exists a homogeneous model 𝒜 with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous 𝒜 with a 0-basis has a d-decidable copy. In previous work, we showed that the nonlow₂ Δ₂⁰ degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding Δ₂⁰ degrees.

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J. Symbolic Logic, Volume 75, Issue 3 (2010), 971-995.

First available in Project Euclid: 9 July 2010

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Lange, Karen. A characterization of the 0 -basis homogeneous bounding degrees. J. Symbolic Logic 75 (2010), no. 3, 971--995. doi:10.2178/jsl/1278682211.

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