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Switch to: Citations
References in:
Bounding Homogeneous Models
Journal of Symbolic Logic 72 (1):305 - 323 (2007)
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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 $\Delta _{0}^{0}$ -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 (...) |
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We say a countable model ������ has a 0-basis if the types realized in ������ are uniformly computable. 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 model ������ with a 0-basis but no decidable copy. We extend this result here. Let d ≤ 0' be any low₂ degree. We show that there exists a homogeneous (...) |
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We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties. If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 0, (...) |
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A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model U of T decidable in X. It is easy to see that $X = 0\prime$ is prime bounding. Denisov claimed that every $X <_{T} 0\prime$ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets $X \leq_{T} 0\prime$ are exactly the sets which are not $low_2$ . Recall that (...) |
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