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Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups

Published online by Cambridge University Press:  12 March 2014

Anatole Khelif
Anatole Khelif*
Affiliation:
UFR de Mathématiques, Equipe de Logique, Université Paris VII, 2, Place Jussieu, 75251 Paris Cedex 05, France, E-mail: khelif@logique.jussieu.fr
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Abstract

We prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).

A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.

Our main result uses constructible sets and can be put in an abstract general framework.

Comparison with the standard methods which use forcing (cf. [4]) is sketched.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 1 , March 1996 , pp. 277 - 284
Copyright
Copyright © Association for Symbolic Logic 1996

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References

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