2018.05.11: Criterion of finite axiomatizability:
a class of models K is finitely axiomatizable iff K and its complement are axiomatizable.
Corollary: the class of infinite groups is axiomatizable, but not finitely axiomatizable;
the class of finite groups is not axiomatizable. The same result holds for any
class of models K that is finitely axiomatizable and has arbitrarily large finite models.
Finite Model Theory: the lack of compactness; the lack of completeness; games still work.
Lemma: Every finite model has a closed formula that characterizes this model up to isomorphism.
Corollary: Two finite models are elementary equivalent iff they are isomorphic.