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Stable theories with a new predicate

Published online by Cambridge University Press:  12 March 2014

Enrique Casanovas  and
Martin Ziegler
Enrique Casanovas
Affiliation:
Departamento de Lógica, Historia y Filosofia de la Ciencia, Universidad de Barcelona, Baldiri Reixac, 08028 Barcelona, Spain, E-mail: casanova@mat.ub.es
Martin Ziegler
Affiliation:
Institut für Mathematische Logik, Albert-Ludwigs Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany, E-mail: ziegler@uni-freiburg.de
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Extract

Let M be an L-structure and A be an infinite subset of M. Two structures can be defined from A:

• The induced structure on A has a name Rφ for every ∅-definable relation φ(M) ∩ An on A. Its language is

A with its Lind-structure will be denoted by Aind.

• The pair (M, A) is an L(P)-structure, where P is a unary predicate for A and L(P) = L ∪{P}.

We call A small if there is a pair (N, B) elementarily equivalent to (M, A) and such that for every finite subset b of N every L–type over Bb is realized in N.

A formula φ(x, y) has the finite cover property (f.c.p) in M if for all natural numbers k there is a set of φ–formulas

which is k–consistent but not consistent in M. M has the f.c.p if some formula has the f.c.p in M. It is well known that unstable structures have the f.c.p. (see [6].) We will prove the following two theorems.

Theorem A. Let A be a small subset of M. If M does not have the finite cover property then, for every λ ≥ ∣L∣, if both M andAindare λ–stable then (M, A) is λ–stable.

Corollary 1.1 (Poizat [5]). If M does not have the finite cover property and N ≺ M is a small elementary substructure, then (M, N) is stable.

Corollary 1.2 (Zilber [7]). If U is the group of wots of unity in the field ℂ of complex numbers the pair (ℂ, U) isω–stable.

Proof. (See [4].) As a strongly minimal set ℂ is ω–stable and does not have the f.c.p. By the subspace theorem of Schmidt [3] every algebraic set intersects U in a finite union of translates of subgroups definable in the group structure of U alone. Whence Uind is nothing more than a (divisible) abelian group, which is ω–stable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1127 - 1140
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

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