Totally Categorical Theories
Theorem: (Zilber; Cherlin, Harrington, Lachlan) If a countable, complete theory is categoricial in every infinite cardinal, then does not have a finite set of axioms. In fact, for such a theory T for any sentence φ ∈ T there is a finite model of φ

The Cherlin-Harrington-Lachlan proof requires the weaker hypothesis of ℵ0-stable, ℵ0-categorical and is based on the classification of the finite simple groups.

Zilber's proof is based on the geometry of strongly minimal sets. He proves a trichotomy theorem for strongly minimal sets, develops an intersection theory for ℵ0-categorical strongly minimal sets and proves theorems analogous to Bezout's theorem, and extends geometrical algebra to the setting of strongly minimal sets so that groups and fields may be recognized from configuarations of depedencies in strongly minimal sets.