Wilkie's Theorem and the Uniform Real Schanuel Conjecture

Abstract

In this thesis we give a detailed proof of the model completeness of two expansions of the real ordered field, specifically the expansion by Pfaffian chains of functions and the expansion by the exponential function. The latter result is also known as Wilkie's Theorem and both of the proofs are due to Alex Wilkie. As an application of Wilkie's Theorem, we provide a modest generalization of the fact that Schanuel's conjecture over the real numbers is equivalent to a uniform version of itself, as proven by Jonathan Kirby and Boris Zilber.