Schanuel's conjecture is a strengthening of the Lindemann-Weierstrass-threorem. It states
If $\lambda_1,\cdots \lambda_n$ are complex numbers linear independent over $\mathbb Q$, then $$\mathbb Q(\lambda_1,\cdots \lambda_n,e^{\lambda_1},\cdots ,e^{\lambda_n})$$ has transcendental degree at least $n$ over $\mathbb Q$
(See http://mathworld.wolfram.com/SchanuelsConjecture.html )
I recently read that in transcendental-theory "anything could be proven", if Schanuel's conjecture would be proven.
One of the consequences would be that $e$ and $\pi$ are algebraically independent. In particular, many combinations of $e$ and $\pi$ could be proven to be transcendental, for example $e+\pi$ , $e\pi$ , $e^e$ and so on. But what could be proven else ?
Can someone give a survey which type of numbers could be proven to be transcendental ? I would like to get an idea of the power of the conjecture and to understand the "anything" better.
Additional questions :
Would Schanuel's conjecture be of any use in the case of the Euler-Mascheroni-constant (I guess no) ?
How strong do mathematicians believe that Schanuel's conjecture is true ? Are there any mathematicians having doubts or do some mathematicians even believe it is false ? (My guess is that the conjecture is strongly believed to be true)