Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups
Anatole Khélif · Journal of Symbolic Logic · 1996
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.