A note on the axioms for Zilber's pseudo-exponential fields

TL;DR

This paper explores the logical properties of Zilber's pseudo-exponential fields, showing their elementary equivalence to complex exponentiation and characterizing their models through first-order types and elementary embeddings.

Contribution

It demonstrates that Zilber's conjecture follows from elementary equivalence and characterizes pseudo-exponential fields as models atomic over exponential transcendence bases.

Findings

Pseudo-exponential fields are models of their common first-order theory.

They are precisely the models atomic over exponential transcendence bases.

The class of pseudo-exponential fields forms a non-finitary abstract elementary class.

Abstract

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 non-finitary abstract elementary class, answering a question of Kes\"al\"a and Baldwin.