On quasiminimal excellent classes

A careful exposition of Zilber’s quasiminimal excellent classes and their categoricity is given, leading to two new results: the Lω1,ω(Q)definability assumption may be dropped, and each class is determined by its model of dimension א0. Boris Zilber developed quasiminimal excellent classes in [Zil05], in order to prove that his conjectural description of complex exponentiation was categorical. This article gives a simplified and careful exposition of quasiminimal excellent classes, and of the categoricity proof. This more careful exposition has led to two new results. In [Zil05], the proof of existence of arbitrarily large models depended on the class being definable by an Lω1,ω(Q)-sentence of a specific form. The question of whether this could be generalized to any Lω1,ω(Q)-sentence was posed. Here we show that the assumption that the class be Lω1,ω(Q)-definable may be dropped entirely (indeed essentially it follows from the other axioms). Although a quasiminimal excellent class may not have arbitrarily large models, it is a subclass of a unique quasiminimal excellent class which does. The second new result is that any quasiminimal excellent class may be produced in a “bootstrap” fashion from its unique model of dimension א0. For precise statements, see theorem 19 and corollaries 20 and 21. Some of the simplifications in the presentation are due to John Baldwin, in particular the construction of the isomorphism in theorem 10 as the union of the maps fX . Other simplifications are due to me. ∗University of Oxford and University of Illinois at Chicago, supported by the EPSRC