On the Real Exponential Field

The thesis is divided into two parts. In the first part, the thesis presents results concerning the model completeness of the theory extending the reals as ordered rings with restricted Pfaffian chains and the theory \(T_{\exp}\) developed by Wilkie. The second part addresses decidability results that follow from Schanuel's Conjecture, proved by Wilkie and Macintyre, and a bound due to Kirby for the number of the size of the essential counterexamples set to Schanuel's Conjecture.