Shepherdson proved that a discrete unitary commutative semi-ring A satisfies IE0 (induction scheme restricted to quantifier free formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings and fields. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory T in the language of rings plus exponentiation such that the models (A, expA) of T are all integral parts A of models M of T with A closed under expM and expA = expM |A. Namely T=EXP , the basic theory of real exponential fields; T=EXP+ the Rolle and the intermediate value properties for all exp-polynomials; and T = Texp, the complete theory of the field of reals with exponentiation.

@inproceedings{BoughattasOnTA,
title={On the arithmetization of real fields with exponentiation},
author={Sedki Boughattas and J-P. Ressayre}
}