A characterization of the 0-basis homogeneous bounding degrees
Karen Lange
Source: J. Symbolic Logic Volume 75, Issue 3
(2010), 971-995.
Abstract
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 d≤0' 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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Zentralblatt MATH identifier: 05795054
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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,Amsterdam, 2000.
Mathematical Reviews (MathSciNet): MR1767842
Zentralblatt MATH: 0960.03001
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 edn. 1973, 2nd edn. 1977.
B.F. Csima, Degree spectra of prime models, Journal of Symbolic Logic, vol. 69 (2004), pp. 430--442.
Mathematical Reviews (MathSciNet): MR2058182
Zentralblatt MATH: 1069.03025
Digital Object Identifier: doi:10.2178/jsl/1082418536
Project Euclid: euclid.jsl/1082418536
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
Zentralblatt MATH: 1116.03027
Digital Object Identifier: doi:10.2178/jsl/1174668397
Project Euclid: euclid.jsl/1174668397
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
Zentralblatt MATH: 1071.03021
Digital Object Identifier: doi:10.2178/jsl/1102022214
Project Euclid: euclid.jsl/1102022214
R. Epstein, Prime models of computably enumerable degree, Journal of Symbolic Logic, vol. 73 (2008), pp. 1373--1388.
Mathematical Reviews (MathSciNet): MR2467224
Zentralblatt MATH: 1155.03017
Digital Object Identifier: doi:10.2178/jsl/1230396926
Project Euclid: euclid.jsl/1230396926
S.S. Goncharov, Strong constructivizability of homogeneous models (Russian), Algebra i Logika, vol. 17 (1978), pp. 363--388, 490, translated in: \em Algebra and Logic, vol. 17 (1978) pp. 247--263.
Mathematical Reviews (MathSciNet): MR538302
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, Elsevier Science, Amsterdam, 1998, pp. 3--114.
Mathematical Reviews (MathSciNet): MR1673621
Zentralblatt MATH: 0952.03037
Digital Object Identifier: doi:10.1016/S0049-237X(98)80002-5
D.R. Hirschfeldt, K.M. Lange, and R.A. Shore, The homogeneous model theorem, in preparation.
D.R. Hirschfeldt, R.A. Shore, and T.A. Slaman, The atomic model theorem and type omitting, Transactions of American Mathematical Society, vol. 361 (2009), pp. 5805--5837.
Mathematical Reviews (MathSciNet): MR2529915
Zentralblatt MATH: 1184.03005
Digital Object Identifier: doi:10.1090/S0002-9947-09-04847-8
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
Zentralblatt MATH: 1099.03024
Digital Object Identifier: doi:10.1090/S0002-9939-05-08097-4
C.G. Jockusch, Jr., Degrees in which the recursive sets are uniformly recursive, Canadian Journal of Mathematics, vol. 24 (1972), pp. 1092--1099.
Mathematical Reviews (MathSciNet): MR321716
J. F. Knight, Degrees coded in jumps of orderings, Journal of Symbolic Logic, vol. 51 (1986), pp. 1034--1042.
Mathematical Reviews (MathSciNet): MR865929
Zentralblatt MATH: 0633.03038
Digital Object Identifier: doi:10.2307/2273915
JSTOR: links.jstor.org
K. M. Lange, The degree spectra of homogeneous models, Journal of Symbolic Logic, vol. 73 (2008), pp. 1009--1028.
Mathematical Reviews (MathSciNet): MR2444283
Zentralblatt MATH: 1160.03014
Digital Object Identifier: doi:10.2178/jsl/1230396762
Project Euclid: euclid.jsl/1230396762
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
Zentralblatt MATH: 1123.03027
Digital Object Identifier: doi:10.1305/ndjfl/1172787551
Project Euclid: euclid.ndjfl/1172787551
D. Marker, Model theory: An introduction, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, 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
Zentralblatt MATH: 0432.03018
Digital Object Identifier: doi:10.1016/0003-4843(78)90030-X
--------, Homogeneous models and decidability, Pacific Journal of Mathematics, vol. 91 (1980), pp. 407--418.
Mathematical Reviews (MathSciNet): MR615688
Zentralblatt MATH: 0467.03007
Project Euclid: euclid.pjm/1102778733
M.G. Peretyat'kin, A criterion for strong constructivizability of a homogeneous model (Russian), Algebra i Logika, vol. 17 (1978), pp. 436--454, 491, translated in: \em 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.
