### The degree spectra of homogeneous models

Karen Lange
Source: J. Symbolic Logic Volume 73, Issue 3 (2008), 1009-1028.

#### 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.

First Page:
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396762
Digital Object Identifier: doi:10.2178/jsl/1230396762
Mathematical Reviews number (MathSciNet): MR2444283
Zentralblatt MATH identifier: 1160.03014

### References

C. J. Ash and J. F. Knight, Computable structures and the hyperarithmetical hierarchy, 1st ed., Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.
Mathematical Reviews (MathSciNet): MR1767842
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, 1990, [1st ed. 1973, 2nd ed. 1977].
B. F. Csima, Degree spectra of prime models, Journal of Symbolic Logic, vol. 69 (2004), pp. 430--442.
Mathematical Reviews (MathSciNet): MR2058182
Digital Object Identifier: doi:10.2178/jsl/1082418536
Project Euclid: euclid.jsl/1082418536
Zentralblatt MATH: 1069.03025
B. F. Csima, V. S. Harizanov, D. R. Hirschfeldt, and R. I. Soare, Bounding homogeneous models, Journal of Symbolic Logic, vol. 72 (2007), pp. 305--323.
Mathematical Reviews (MathSciNet): MR2298484
Digital Object Identifier: doi:10.2178/jsl/1174668397
Project Euclid: euclid.jsl/1174668397
Zentralblatt MATH: 1116.03027
B. F. Csima, D. R. Hirschfeldt, J. F. Knight, and R. I. Soare, Bounding prime models, Journal of Symbolic Logic, vol. 69 (2004), pp. 1117--1142.
Mathematical Reviews (MathSciNet): MR2135658
Digital Object Identifier: doi:10.2178/jsl/1102022214
Project Euclid: euclid.jsl/1102022214
Zentralblatt MATH: 1071.03021
R. Epstein, Computably enumerable degrees of Vaught's models, submitted.
S. S. Goncharov, Strong constructivizability of homogeneous models, Algebra i Logika, vol. 17 (1978), pp. 363--388, 490, in Russian [translated in: Algebra and Logic, vol. 17 (1978), pp. 247--263].
Mathematical Reviews (MathSciNet): MR538302
S. S. Goncharov and A. T. Nurtazin, Constructive models of complete decidable theories, Algebra i Logika, vol. 12 (1973), pp. 125--142, 243, [translated in: Algebra and Logic, vol. 12 (1973), pp. 67--77].
Mathematical Reviews (MathSciNet): MR398816
V. S. Harizanov, Pure computable model theory, Handbook of recursive mathematics (Yu. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel, editors), Studies in Logic and the Foundations of Mathematics, vol. 138--139, North-Holland, Amsterdam, 1998, pp. 3--114.
Mathematical Reviews (MathSciNet): MR1673621
Digital Object Identifier: doi:10.1016/S0049-237X(98)80002-5
L. Harrington, Recursively presentable prime models, Journal of Symbolic Logic, vol. 39 (1974), pp. 305--309.
Mathematical Reviews (MathSciNet): MR351804
Digital Object Identifier: doi:10.2307/2272643
Zentralblatt MATH: 0332.02055
K. Harris, Bounding saturated models, in preparation.
D. R. Hirshfeldt, Computable trees, prime models, and relative decidability, Proceedings of the American Mathematical Society, vol. 134, 2006, pp. 1495--1498.
Mathematical Reviews (MathSciNet): MR2199197
Digital Object Identifier: doi:10.1090/S0002-9939-05-08097-4
Zentralblatt MATH: 1099.03024
D. R. Hirschfeldt, R. A. Shore, and T. A. Slaman, The atomic model theorem, submitted.
J. F. Knight, Degrees coded in jumps of orderings, Journal of Symbolic Logic, vol. 51 (1986), pp. 1034--1042.
Mathematical Reviews (MathSciNet): MR865929
Digital Object Identifier: doi:10.2307/2273915
Zentralblatt MATH: 0633.03038
K. M. Lange, A characterization of the $\0$-basis homogeneous bounding degrees, in preparation.
--------, Reverse mathematics of homogeneous models, in preparation.
K. M. Lange and R. I. Soare, Computability of homogeneous models, Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 143--170.
Mathematical Reviews (MathSciNet): MR2289903
Digital Object Identifier: doi:10.1305/ndjfl/1172787551
Project Euclid: euclid.ndjfl/1172787551
Zentralblatt MATH: 1123.03027
D. Marker, Model theory: an introduction, Graduate Texts in Mathematics, vol. 277, New York, 2002.
Mathematical Reviews (MathSciNet): MR1924282
T. S. Millar, Foundations of recursive model theory, Annals of Mathematical Logic, vol. 13 (1978), pp. 45--72.
Mathematical Reviews (MathSciNet): MR482430
Digital Object Identifier: doi:10.1016/0003-4843(78)90030-X
Zentralblatt MATH: 0432.03018
T. S. Millar, Homogeneous models and decidability, Pacific Journal of Mathematics, vol. 91 (1980), pp. 407--418.
Mathematical Reviews (MathSciNet): MR615688
Project Euclid: euclid.pjm/1102778733
--------, Type structure complexity and decidability, Transactions of the American Mathematical Society, vol. 271 (1982), pp. 73--81.
Mathematical Reviews (MathSciNet): MR648078
Digital Object Identifier: doi:10.2307/1998751
Zentralblatt MATH: 0493.03010
M. Morley, Decidable models, Israel Journal of Mathematics, vol. 25 (1976), pp. 233--240.
Mathematical Reviews (MathSciNet): MR457190
Digital Object Identifier: doi:10.1007/BF02757002
Zentralblatt MATH: 0361.02067
M. G. Peretyat'kin, A criterion for strong constructivizability of a homogeneous model, Algebra i Logika, vol. 17 (1978), pp. 436--454, 491, in Russian, [translated in: Algebra and Logic, vol. 19 (1980), pp. 202--229].
Mathematical Reviews (MathSciNet): MR538306
R. I. Soare, Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.
Mathematical Reviews (MathSciNet): MR882921
--------, Computability theory and applications, Springer-Verlag, Heidelberg, in preparation.
D. A. Tusupov, Numerations of homogeneous models of decidable complete theories with a computable family of types, Theory of algorithms and its applications (V. N. Remeslennikov, editor), Computable Systems, vol. 129.
R. L. Vaught, Denumerable models of complete theories, Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959), Pergamon Press, 1961, pp. 301--321.
Mathematical Reviews (MathSciNet): MR186552