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The classification of small weakly minimal sets. II

Published online by Cambridge University Press:  12 March 2014

Steven Buechler
Steven Buechler*
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
*
Department of Mathematics, University of Wisconsin at Milwaukee, Milwaukee, Wisconsin 53201
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Abstract

The main result is Vaught's conjecture for weakly minimal, locally modular and non-ω-stable theories. The more general results yielding this are the following.

Theorem A. Suppose that T is a small unidimensional theory and D is a weakly minimal set, definable over the finite set B. Then for all finite A ⊂ D there are only finitely many nonalgebraic strong types over B realized in acl(A) ∩ D.

Theorem B. Suppose that T is a small, unidimensional, non-ω-stable theory such that the universe is weakly minimal and locally modular. Then for all finite A there is a finite B ⊂ cl(A) such that a ∈ cl(A) iff a ∈ cl(b) for some b ∈ B.

Recall the property (S) defined in the abstract of [B1].

Theorem C. Let T be as in Theorem B. Then, if T does not satisfy (S), T has many countable models.

Combining Theorem C and the results in [B1] we obtain Vaught's conjecture for such theories.

Type
Survey/expository papers
Information
The Journal of Symbolic Logic , Volume 53 , Issue 2 , June 1988 , pp. 625 - 635
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[B1] Buechler, S., The classification of small weakly minimal sets. I, Classification theory (Chicago, Illinois, 1985), Springer-Verlag, Berlin (to appear).Google Scholar
[B2] Buechler, S., The geometry of weakly minimal types, this Journal, vol. 50 (1985), pp. 1044–1053.Google Scholar
[B3] Buechler, S., “Geometrical” stability theory, Logic Colloquium '85, North-Holland, Amsterdam, 1987, pp. 53–66.Google Scholar
[Hr] Hrushovski, E., Contributions to stable model theory, Ph.D. thesis, University of California, Berkeley, California, 1986.Google Scholar
[L] Loveys, J., Certain weakly minimal theories, Ph.D. thesis, Simon Fraser University, Burnaby, British Columbia, 1986.Google Scholar
[S] Saffe, J., On Vaught's conjecture for superstable theories, preprint, 1982.Google Scholar