Forcing the truth of a weak form of Schanuel’s conjecture

Schanuel’s conjecture states that the transcendence degree over $\mathbb{Q}$ of the $2n$-tuple $(\lambda _1,\dots ,\lambda _n,e^{\lambda _1},\dots ,e^{\lambda _n})$ is at least $n$ for all $\lambda _1,\dots ,\lambda _n\in \mathbb{C}$ which are linearly independent over $\mathbb{Q}$; if true it would settle a great number of elementary open problems in number theory, among which the transcendence of $e$ over $\pi $. Wilkie [11], and Kirby [4, Theorem 1.2] have proved that there exists a smallest countable algebraically and exponentially closed subfield $K$ of $\mathbb{C}$ such that Schanuel’s conjecture holds relative to $K$ (i.e. modulo the trivial counterexamples, $\mathbb{Q}$ can be replaced by $K$ in the statement of Schanuel’s conjecture). We prove a slightly weaker result (i.e. that there exists such a countable field $K$ without specifying that there is a smallest such) using the forcing method and Shoenfield’s absoluteness theorem. This result suggests that forcing can be a useful tool to prove theorems (rather than independence results) and to tackle problems in domains which are apparently quite far apart from set theory.