A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L ω₁ ,ω (Q)-definability assumption may be dropped, and each class is determined by its model of dimension $\aleph _{0}$.

We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary (...) class, answering a question of Kesälä and Baldwin. (shrink)

Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.