Abstract
In this paper we obtain two results using amalgamation classes and Fraïssé limits. First, we construct a decidable theory T whose types are all decidable yet whose prime model is not decidable. Millar [15] constructed such example but his example uses an infinite language in an essential way. Our example uses one binary predicate symbol, that is, the models we construct are graphs. Second, for any finite lattice \(\cal F\) we construct a theory T with countably many models such that the fundamental order determined by T is isomorphic to \(\cal F\). As a by-product of this example, we propose the investigation of computable and decidable models of T by connecting them to the fundamental order of T.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baldwin, J., Berman, J.: Concrete representations of lattices and the fundamental order. Classification Theory, 24–31 (1987)
Csima, B., Kalimullin, I.: Degree spectra and immunity properties. Mathematical Logic Quarterly 56(1), 67–77 (2010)
Gavruskin, A.: Computable limit models. In: Programs, Proofs, Processes—CiE, pp. 188–193 (2010)
Goncharov, S.: Strong constructivizability of homogeneous models. Algebra and Logic 17(4), 247–263 (1978)
Goncharov, S., Ershov, Y.: Constructive Models. Consultants Bureau, New York (2000)
Goncharov, S., Nurtazin, A.: Constructive models of complete solvable theories. Algebra and Logic 12(2), 67–77 (1973)
Greenberg, N., Montalbán, A., Slaman, T.: Relative to any non-hyperarithmetic set. preprint arXiv:1110.1907 (2011)
Harrington, L.: Recursively presentable prime models. The Journal of Symbolic Logic 39(2), 305–309 (1974)
Hirschfeldt, D.: Computable trees, prime models, and relative decidability. Proceedings of the American Mathematical Society 134(5), 1495–1498 (2006)
Hodges, W.: Model Theory. In: Encyclopaedia of Mathematics and Its Applications, vol. 42, Cambridge University Press (1993)
Khoussainov, B., Semukhin, P., Stephan, F.: Applications of Kolmogorov complexity to computable model theory. The Journal of Symbolic Logic 72(3), 1041–1054 (2007)
Khoussainov, B., Nies, A., Shore, R.: Computable models of theories with few models. Notre Dame Journal of Formal Logic 38(2), 165–178 (1997)
Lange, K.: The degree spectra of homogeneous models. Journal of Symbolic Logic 73(3), 1009–1028 (2008)
Lascar, D., Poizat, B.: An introduction to forking. The Journal of Symbolic Logic 44(3), 330–350 (1979)
Millar, T.: Foundations of recursive model theory. Annals of Mathematical Logic 13, 45–72 (1978)
Millar, T.: Omitting types, type spectrums, and decidability. Journal of Symbolic Logic 48(1), 171–181 (1983)
Morley, M.: Decidable Models. Israel Journal of Mathematics 25, 233–240 (1976)
Peretyatkin, M.: On complete theories with a finite number of denumerable models. Algebra and Logic 12(5), 310–326 (1973)
Peretyatkin, M.: Criterion for strong constructivizability of a homogeneous model. Algebra and Logic 17(4), 290–301 (1978)
Poizat, B.: Attention a la Marche! Journal of Symbolic Logic 51(3), 570–585 (1986)
Sacks, G.: Saturated model theory, 2nd edn. World Scientific Publishing Company Incorporated (2010)
Sudoplatov, S.: Complete theories with finitely many countable models I, II. Algebra and Logic 43(1), 62–69 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gavruskin, A., Khoussainov, B. (2013). On Decidable and Computable Models of Theories. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-39053-1_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39052-4
Online ISBN: 978-3-642-39053-1
eBook Packages: Computer ScienceComputer Science (R0)