Notre Dame Journal of Formal Logic

A Note on the Axioms for Zilber’s Pseudo-Exponential Fields

Jonathan Kirby

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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.

Article information

Notre Dame J. Formal Logic, Volume 54, Number 3-4 (2013), 509-520.

First available in Project Euclid: 9 August 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C65: Models of other mathematical theories
Secondary: 03C48: Abstract elementary classes and related topics [See also 03C45]

pseudo-exponentiation exponential fields Schanuel property first-order theory abstract elementary class


Kirby, Jonathan. A Note on the Axioms for Zilber’s Pseudo-Exponential Fields. Notre Dame J. Formal Logic 54 (2013), no. 3-4, 509--520. doi:10.1215/00294527-2143844.

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