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A note on the axioms for Zilber's pseudo-exponential fields
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.Comment: 10 pages, v2: substantial alteration