X-Nico

Pfaffian function

In the 1990s, Alex Wilkie showed that one has the same result if instead of adding every analytic function, one just adds the exponential function to R to get the ordered real field with exponentiation, R_exp, a result known as Wilkie's theorem.

Transcendental function

In fact, Alex Wilkie showed that the situation is even worse: he constructed a transcendental function ƒ: R → R that is analytic everywhere but whose transcendence cannot be detected by any first-order method.

see also