Notes on quasiminimality and excellence

Zilber proposes [62] to prove ‘canonicity results for pseudo-analytic’ structures. Informally, ‘canonical’ means ‘the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power’ while ‘pseudoanalytic’ means ‘the model of power 2א0 can be taken as a reduct of an expansion of the complex numbers by analytic functions’. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [32], this line was taken up in a long series of works by Shelah. We place Zilber’s work in this context. The second direction stems from Hrushovski’s construction of a counterexample to Zilber’s conjecture that every strongly minimal set is ‘trivial’, ‘vector space-like’, or ‘field-like’. This construction turns out to be a very concrete example of an Abstract Elementary Class, a concept that arose in Shelah’s analysis. This paper examines the intertwining of these three themes. For simplicity, we work in a countable vocabulary.