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

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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 non-finitary abstract elementary class, answering a question of Kes\"al\"a and Baldwin.
Original languageEnglish
Pages (from-to)509-520
Number of pages12
JournalNotre Dame Journal of Formal Logic
Issue number3-4
Publication statusPublished - 2013


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

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