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

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 Kesälä and Baldwin.

10 pages, v2: substantial alterations

References in corpus (4)

• On Quasiminimal Excellent Classes
• Exponential algebraicity in exponential fields
• Exponentially Closed Fields and the Conjecture on Intersections with Tori
• Finitely Presented Exponential Fields

Cited by in corpus (7)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Exponentially Closed Fields and the Conjecture on Intersections with Tori
• Abstract elementary classes stable in ${\aleph }_0$
• The algebraic numbers definable in various exponential fields
• Structural Logic and Abstract Elementary Classes with Intersection
• A pseudoexponentiation-like structure on the algebraic numbers
• Algebraic types in Zilber's exponential field