Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups
Copyright policy )Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups
AbstractWe 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.
- University of Paris France
- Université Paris Cité France
elementary equivalence, Inner models, including constructibility, ordinal definability, and core models, ordered groups, model companion, ordered \(k\)-algebras, Model-theoretic algebra, Ordered groups, ordered field, existentially closed models, axiom of constructibility
elementary equivalence, Inner models, including constructibility, ordinal definability, and core models, ordered groups, model companion, ordered \(k\)-algebras, Model-theoretic algebra, Ordered groups, ordered field, existentially closed models, axiom of constructibility
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