Schanuel's conjecture and abstract elementary classes 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?
 A: 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:


*

*$F$ is an algebraically closed field of  characteristic zero

*$\forall x\forall y [\exp(x + y) = \exp(x) \cdot \exp(y)]$

*$\mathop{\rm ker}(\exp)=2\pi i\mathbb{Z}$

*the class is existentially closed

*the class has the countable closure property, i.e. every analytic subset of $F^n$ of dimension $0$ is essentially countable, and

*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 $e^x$. 
