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.
"A Note on the Axioms for Zilber’s Pseudo-Exponential Fields." Notre Dame J. Formal Logic 54 (3-4) 509 - 520, 2013. https://doi.org/10.1215/00294527-2143844