### Computable Models of Theories with Few Models

Bakhadyr Khoussainov, Andre Nies, and Richard A. Shore
Source: Notre Dame J. Formal Logic Volume 38, Number 2 (1997), 165-178.

#### Abstract

In this paper we investigate computable models of -categorical theories and Ehrenfeucht theories. For instance, we give an example of an -categorical but not -categorical theory such that all the countable models of except its prime model have computable presentations. We also show that there exists an -categorical but not -categorical theory such that all the countable models of except the saturated model, have computable presentations.

First Page:
Primary Subjects: 03C15
Secondary Subjects: 03C35, 03C57
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1039724885
Mathematical Reviews number (MathSciNet): MR1489408
Digital Object Identifier: doi:10.1305/ndjfl/1039724885
Zentralblatt MATH identifier: 0891.03013

### References

[1] Baldwin, J., and A. Lachlan, On strongly minimal sets,'' The Journal of Symbolic Logic, vol. 36 (1971), pp. 79--96.
Mathematical Reviews (MathSciNet): MR44:3851
Zentralblatt MATH: 0217.30402
Digital Object Identifier: doi:10.2307/2271517
[2] Ershov, Yu., Constructive Models and Problems of Decidability, Nauka, Moskow, 1980.
Mathematical Reviews (MathSciNet): MR598465
[3] Goncharov, S., Constructive models of $\omega_1$-categorical theories,'' Matematicheskie Zametki, vol. 23 (1978), pp. 885--9.
Mathematical Reviews (MathSciNet): MR80g:03029
Zentralblatt MATH: 0403.03025
[4] Goncharov, S., Strong constructivability of homogeneous models,'' Algebra and Logic, vol. 17 (1978), pp. 363--8.
Mathematical Reviews (MathSciNet): MR538302
[5] Logic Notebook, edited by Yu. Ershov and S. Goncharov, Novosibirsk University, Novosibirsk, 1986.
Mathematical Reviews (MathSciNet): MR88k:03003
[6] Khissamiev, N.,On strongly constructive models of decidable theories,'' Izvestiya AN Kaz. SSR, vol. 1 (1974), pp. 83--4.
Mathematical Reviews (MathSciNet): MR354344
[7] Kudeiberganov, K., On constructive models of undecidable theories,'' Siberian Mathematical Journal, vol. 21 (1980), pp. 155--8.
Mathematical Reviews (MathSciNet): MR592228
[8] Harrington, L., Recursively presentable prime models,'' The Journal of Symbolic Logic, vol. 39 (1973), pp. 305--9.
Mathematical Reviews (MathSciNet): MR50:4292
Zentralblatt MATH: 0332.02055
Digital Object Identifier: doi:10.2307/2272643
[9] Millar, T., The theory of recursively presented models,'' Ph.D. Dissertation, Cornell University, Ithaca, 1976.
[10] Morley, M., Decidable models,'' Israel Journal of Mathematics, vol. 25 (1976), pp. 233--40.
Mathematical Reviews (MathSciNet): MR56:15405
Zentralblatt MATH: 0361.02067
Digital Object Identifier: doi:10.1007/BF02757002
[11] Peretýatkin, M., On complete theories with finite number of countable models,'' Algebra and Logic, vol. 12 (1973), pp. 550--70.
Zentralblatt MATH: 0298.02047
Mathematical Reviews (MathSciNet): MR354347
[12] Rosenstein, J., Linear Orderings, Academic Press, New York, 1982.
Mathematical Reviews (MathSciNet): MR84m:06001
Zentralblatt MATH: 0488.04002