The degree spectra of homogeneous models. (English) Zbl 1160.03014

Summary: 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 \(\mathcal A\) has a \({\mathbf d}\)-basis if the types realized in \(\mathcal A\) are all computable and the Turing degree \({\mathbf d}\) can list \(\Delta _{0}^{0}\)-indices for all types realized in \(\mathcal A\). We say \(\mathcal A\) has a \({\mathbf d}\)-decidable copy if there exists a model \(\mathcal B\cong \mathcal A\) such that the elementary diagram of \(\mathcal B\) is \({\mathbf d}\)-computable. Goncharov, Millar, and Peretyat’kin independently showed there exists a homogeneous \(\mathcal A\) with a \({\mathbf 0}\)-basis but no decidable copy.
We prove that any homogeneous \(\mathcal A\) with a \({\mathbf 0}'\)-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 \(\mathcal A\) is a homogeneous model with a \({\mathbf 0}\)-basis, we show \(\mathcal A\) has copies decidable in every nonzero degree. A degree \({\mathbf d}\) is \({\mathbf 0}\)-homogeneous bounding if any automorphically nontrivial homogeneous \(\mathcal A\) with a \({\mathbf 0}\)-basis has a \({\mathbf d}\)-decidable copy. We show that the nonlow\(_{2}\) \(\Delta _{2}^{0}\) degrees are \({\mathbf 0}\)-homogeneous bounding.


03C57 Computable structure theory, computable model theory
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[1] DOI: 10.1007/BF02757002 · Zbl 0361.02067
[2] Algebra i Logika 17 pp 491– (1978)
[3] DOI: 10.2140/pjm.1980.91.407 · Zbl 0467.03007
[4] Handbook of recursive mathematics 138–139 pp 3– (1998)
[5] Algebra i Logika 12 pp 243– (1973)
[6] Algebra i Logika 17 pp 490– (1978)
[7] Bounding prime models 69 pp 1117– (2004)
[8] Bounding homogeneous models 72 pp 305– (2007)
[9] Degree spectra of prime models 69 pp 430– (2004)
[10] Model theory 73 (1990)
[11] Computable structures and the hyperarithmetical hierarchy 144 (2000) · Zbl 0960.03001
[12] DOI: 10.1016/0003-4843(78)90030-X · Zbl 0432.03018
[13] Model theory: an introduction 277 (2002)
[14] DOI: 10.1305/ndjfl/1172787551 · Zbl 1123.03027
[15] Degrees coded in jumps of orderings 51 pp 1034– (1986) · Zbl 0633.03038
[16] Proceedings of the American Mathematical Society 134 pp 1495– (2006)
[17] Recursively presentable prime models 39 pp 305– (1974)
[18] Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959) pp 301– (1961)
[19] Theory of algorithms and its applications 129
[20] Computability theory and applications · Zbl 0783.68002
[21] Recursively enumerable sets and degrees: A study of computable functions and computably generated sets (1987) · Zbl 0667.03030
[22] DOI: 10.1090/S0002-9947-1982-0648078-8
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