TITLE: Axiomatizations of the real exponential field
QUESTION [17 upvotes]: According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an axiomatization might look like. Has any progress been made on this since  the early 2000s when Marker wrote this?

REPLY [7 votes]: In her PhD thesis ``On the First Order Theory of Real Exponentiation'', Tamara Servi, has given a recursive subtheory
$T$ of $T_{exp}$ and has shown that if Schanuel's conjecture holds, then $T$ is complete and hence provides a recursive axiomatization of $T_{exp}$. There are some related results.
See section 4.7 of the above mentioned paper.