We prove that there are $2^{\chi 0}$ 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.