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Finite extensions and the number of countable models

Published online by Cambridge University Press:  12 March 2014

Terrence Millar
Terrence Millar*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
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Abstract

An Ehrenfeucht theory is a complete first order theory with exactly n countable models up to isomorphism, 1 < n < ω. Numerous results have emerged regarding these theories ([1]–[15]). A general question in model theory is whether or not the number of countable models of a complete theory can be different than the number of countable models of a complete consistent extension of the theory by finitely many constant symbols. Examples are known of Ehrenfeucht theories that have complete extensions by finitely many constant symbols such that the extensions fail to be Ehrenfeucht ([4], [8], [13]). These examples are easily modified to allow finite increases in the number of countable models.

This paper contains examples in the other direction—complete theories that have consistent extensions by finitely many constant symbols such that the extensions have fewer countable models. This answers affirmatively a question raised by, among others, Peretyat'kin [8]. The first example will be an Ehrenfeucht theory with exactly four countable models with an extension by a constant symbol that has only three countable models. The second example will be a complete theory that is not Ehrenfeucht, but which has an extension by a constant symbol that is Ehrenfeucht. The notational conventions for this paper are standard.

Peretyat'kin introduced the theory of a dense binary branching tree with a meet operator [7]. Dense ω-branching trees have also proven useful [5], [11]. Both of the Theories that will be constructed make use of dense ω-branching trees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 264 - 270
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1] Ash, C. J. and Millar, T. S., Persistently finite, persistently arithmetic theories, Proceedings of the American Mathematical Society, vol. 89 (1983), pp. 487–492.CrossRefGoogle Scholar
[2] Benda, M., Remarks on countable models, Fundamenta Mathematicae, vol. 81 (1974). pp. 107–119.CrossRefGoogle Scholar
[3] Lachlan, A. H., On the number of countable models of a countable superstable theory, Logic, methodology and philosophy of science. IV (Proceedings, Bucharest, 1971), North-Holland, Amsterdam, 1973, pp. 45–56.Google Scholar
[4] Millar, T. S., Stability, complete extensions, and the number of countable models, Aspects of effective algebra (Crossley, J. N., editor), Upside Down A Book Company, Yarra Glen, 1981, pp. 196–205.Google Scholar
[5] Millar, T. S., Persistently finite theories with hyperarithmetic models, Transactions of the American Mathematical Society, vol. 278 (1983), pp. 91–99.CrossRefGoogle Scholar
[6] Millar, T. S., Decidable Ehrenfeucht theories, Recursion theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 311–321.CrossRefGoogle Scholar
[7] Peretyat'kin, M. G., On complete theories with a finite number of countable models, Algebra and Logic, vol. 12 (1973), pp. 310–326.CrossRefGoogle Scholar
[8] Peretyat'kin, M. G., A theory with three countable models, Algebra and Logic, vol. 19 (1980), pp. 139–147.CrossRefGoogle Scholar
[9] Pillay, A., Number of countable models, this Journal, vol. 43 (1978), pp. 492–496.Google Scholar
[10] Pillay, A., Instability and theories with few models, Proceedings of the American Mathematical Society, vol. 80 (1980), pp. 461–468.CrossRefGoogle Scholar
[11] Reed, G R., A decidable Ehrenfeucht theory with exactly two hyperarithmetic models, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin, 1986.Google Scholar
[12] Tsuboi, A., On theories having a finite number of nonisomorphic countable models, this Journal, vol. 50(1985), pp. 806–808.Google Scholar
[13] Woodrow, R. E., A note on countable complete theories having three isomorphism types of countable models, this Journal, vol. 41 (1976), pp. 492–496.Google Scholar
[14] Woodrow, R. E., Theories with a finite number of countable models, this Journal, vol. 43 (1978), pp. 442–455.Google Scholar
[15] Vaught, R. L., Denumerable models of complete theories, Infinitistic methods (proceedings, Warsaw, 1959), PWN, Warsaw, and Pergamon Press, Oxford, 1961, pp. 303–321.Google Scholar