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Vaught's conjecture for o-minimal theories

Published online by Cambridge University Press:  12 March 2014

Laura L. Mayer*
Affiliation:
Beloit College, Beloit, Wisconsin 53511

Extract

The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory.

We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2 α , then no structure theorem is expected (cf. [Sh1]).

O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set.

In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension.

In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[Bo] Bouscaren, E., Martin's conjecture for ω-stable theories, Israel Journal of Mathematics, vol. 49 (1984), pp. 1525.Google Scholar
[D] Van Den Dries, L., Remarks on Tarski's problem concerning (R, +,·, exp), Logic Colloquium ’82 (Lolli, G. et al., editors), North-Holland, Amsterdam, 1984, pp. 97121.CrossRefGoogle Scholar
[KPS] Knight, J., Pillay, A., and Steinhorn, C., Definable sets in ordered structures. II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593605.CrossRefGoogle Scholar
[L] Lachlan, A. H., The number of countable models of a countable superstable theory, Logic, methodology and philosophy of science. IV ( Proceedings, Bucharest, 1971 ; Suppes, P. et al., editors), North-Holland, Amsterdam, 1973, pp. 4556.Google Scholar
[Ml] Marker, D., Omitting types in -minimal theories, this Journal, vol. 51 (1986), pp. 6374.Google Scholar
[M2] Marker, D., Vaught's conjecture for somewhat discrete -minimal theories, handwritten notes, University of California, Berkeley, California, 1986.Google Scholar
[Mi] Miller, A. W., Review of [Mo], this Journal, vol. 49 (1984), pp. 314315.Google Scholar
[Mo] Morley, M., The number of countable models, this Journal, vol. 35 (1970), pp. 1418.Google Scholar
[P] Pillay, A., Countable models of stable theories, preprint.Google Scholar
[PS] Pillay, A. and Steinhorn, C., Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol 295 (1986), pp. 565592.CrossRefGoogle Scholar
[R] Rosenstein, J. G., Linear orderings, Academic Press, New York, 1982.Google Scholar
[Ru] Rubin, M., Theories of linear order, Israel Journal of Mathematics, vol. 17 (1974), pp. 392443.Google Scholar
[Sh1] Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[Sh2] Shelah, S., End extensions and numbers of countable models, this Journal, vol. 43 (1978), pp. 550562.Google Scholar
[V] Vaught, R., Denumerable models of complete theories, Infinitistic methods (Proceedings, Warsaw, 1959), PWN, Warsaw, and Pergamon Press, Oxford, 1961, pp. 303321.Google Scholar
[Wl] Wagner, C. M., Martin's conjecture for trees, Abstracts of Papers Presented to the American Mathematical Society, vol. 2 (1981), p. 528 (abstract 81T-03-549).Google Scholar
[W2] Wagner, C. M., On Martin's conjecture, Annals of Mathematical Logic, vol. 22(1982), pp. 4767.CrossRefGoogle Scholar