Angus Macintyre

  Exponential-algebraic real numbers, Schanuel's Conjecture, and The
	    Decision Problem for Real Exponentiation


A.J.Wilkie and I proved that there is an algorithm for deciding all
first-order statements about the real exponential field, provided
Schanuel's Conjecture in transcendence theory is true. In our proof we
make essential use of an exponential subfield of the reals, the field
of exponential-algebraic real numbers, which stands to the real
exponential field as the real algebraic numbers stand to the real
field. The definition of these numbers is not straightforward, and
many questions about them remain open. I will discuss the issue of the
exponential algebraicity of pi, and the right notion of exponential
algebraic function.