Constructing ω-stable structures: model completeness Original Research Article

The projective plane of Baldwin (Amer. Math. Soc. 342 (1994) 695) is model complete in a language with additional constant symbols. The infinite rank bicolored field of Poizat (J. Symbolic Logic 64 (1999) 1339) is not model complete. The finite rank bicolored fields of Baldwin and Holland (J. Symbolic Logic 65 (2000) 371; Notre Dame J. Formal Logic (2001), to appear) are model complete. More generally, the finite rank expansions of a strongly minimal set obtained by adding a ‘random’ unary predicate are almost strongly minimal and model complete provided the strongly minimal set is ‘well-behaved’ and admits ‘exactly rank k formulas’. The last notion is a geometric condition on strongly minimal sets formalized in this paper.