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On theories having a finite number of nonisomorphic countable models

Published online by Cambridge University Press:  12 March 2014

Akito Tsuboi
Akito Tsuboi*
Affiliation:
Institute of Mathematics, University of Tsukuba, Sakura-Mura, Niihari-Gun Ibaraki, 305, Japan
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Extract

In this paper we shall state some interesting facts concerning non-ω-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from ω-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory.

Many papers including [4] and [5] are motivated by the conjecture that every non-ω-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory T0 of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory T0 is stable and ω-categorical, then T0 cannot be extended to a theory T which has n countable models (1 < n < ω) by adding axioms for constant symbols.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985 , pp. 806 - 808
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

[1]Benda, M., Remarks on countable models, Fundamenta Mathematical vol. 81 (1974), pp. 107–119.CrossRefGoogle Scholar
[2]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[3]Pillay, A., Number of countable models, this Journal, vol. 43 (1978), pp. 492–496.Google Scholar
[4]Pillay, A., Instability and theories with few models, Proceedings of the American Mathematical Society, vol. 80 (1980), pp. 461–468.CrossRefGoogle Scholar
[5]Pillay, A., Stable theories, pseudoplanes and the number of countable models (to appear).Google Scholar
[6]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[7]Woodrow, R. E., A note on countable models, this Journal, vol. 41 (1976), pp. 672–680.Google Scholar
[8]Woodrow, R. E., Theories with a finite number of countable models, this Journal, vol. 43 (1978), pp. 442–455.Google Scholar