According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but noone 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?
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. 


protected by Felipe Voloch Aug 9 '15 at 23:40
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