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

Abstract

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.