Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-02T23:03:10.839Z Has data issue: false hasContentIssue false

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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