Zilber's trichotomy theorem
Theorem: (Zilber) If X is a strongly minimal set, then exactly one of the following
is true X.
- X is trivial in the sense that acl defines a degenerate
pregeometry on X (and any elementary extension):
acl(A) = ∪ { acl({a}) | a ∈ A } for all
A ⊂ X
- X is essentially a vector space (formally, nontrivial,
locally modular): That is, possibly after adding
some constant symbols to the language of X, there is an infinite group
space G bi-interpretable with X for
which every definable subset of any Cartesian power of G is
a finite Boolean combination of cosets of definable subgroups.
- X is "field-like:" there is a type definable
pseudoplane interpretable in X.