Of course the details are very tricky :) "behaves like C" means that you construct a first-order theory that all the properties of C that you want, plus the con

Of course the details are very tricky :) "behaves like C" means that you construct a first-order theory that all the properties of C that you want, plus the conjecture. I don't know much about this, but I think that there is no conjugation in the language of exponential fields, and if you try to add it everything becomes intractable (for one, every proof would first require to prove Schanuel's conjecture). The Wikipedia has a few references: https://en.wikipedia.org/wiki/Schanuel%27s_conjecture#Zilber.27s_pseudo-exponentiation https://en.wikipedia.org/wiki/Schanuel%27s_conjecture#Zilber...