Schanuel's conjecture states:

- If $\alpha_1,\alpha_2,...,\alpha_n$ are complex numbers linearly independent over $\mathbb{Q}$, then the transcendence degree of the field $\mathbb{Q}(\alpha_1,e^{\alpha_1},\alpha_2,e^{\alpha_2},...,\alpha_n,e^{\alpha_n})$ over $\mathbb{Q}$ is at least $n$.

In What is a closed-form number (The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448; MSN), Timothy Y. Chow states the following conjecture:

A) The real root $R$ of the equation $x+e^x=0$ is not in $\mathbb{E}$ (having defined $\mathbb{E}$ as the field of all ‘logarithmic exponential numbers’, which is the intersection of all subfields in $\mathbb{C}$ that are closed under $\exp$ and $\log$).

He also states:

- Schanuel's conjecture implies conjecture A).

My question is: if Schanuel's conjecture isn't true, does this mean we can find the solution $R$ of $x+e^x=0$ with $R\in\mathbb{E}$?

$K\subset\mathbf{C}$ is closed under $\log$" means "$x\in\mathbf{C}$, $\exp(x)\in K$ $\Rightarrow$ $x\in K$"? or means the a priori weaker "$y\in K$ $\Rightarrow$ $\exists x\in K:\exp(x)=y$"? $\endgroup$ – YCor Jul 7 at 10:27