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Are there any connections between Schanuel's conjecture and abstract elementary classes. More precisely

Question. Is there any conjecture in abstract elementary classes whose truth implies the Schanuel's conjecture?

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This is an old question, but in case you are still interested the work of Zilber is the most relevant to your question. I am copying and pasting from Rami Grossberg's "Classification Theory for AECs".

Let $K_{\exp}$ consist of all triples of the form $\langle F, +, \cdot, \exp \rangle$ that satisfy the following:

  1. $F$ is an algebraically closed field of characteristic zero
  2. $\forall x\forall y [\exp(x + y) = \exp(x) \cdot \exp(y)]$
  3. $\mathop{\rm ker}(\exp)=2\pi i\mathbb{Z}$
  4. the class is existentially closed
  5. the class has the countable closure property, i.e. every analytic subset of $F^n$ of dimension $0$ is essentially countable, and
  6. the class satisfies Schanuel's Conjecture.

Then Zilber proved that $K_{\exp}$ is $\aleph_1$-categorical and excellent. By results of Shelah $K_{\exp}$ has arbitrarily large models and it is categorical in every uncountable cardinality. This implies that $K_{\exp}$ has a unique model of size $2^{\aleph_0}$. So, Zilber concluded that in order to prove Schanuel's conjecture it suffices to prove that the function $\exp(x)$ defined on $\mathbb{C}$ is indeed $e^x$.

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  • $\begingroup$ @მამუკაჯიბლაძე Yes, you are right. I will edit. $\endgroup$ Jan 25, 2018 at 10:33

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