Schanuel’s conjecture states that the transcendence degree over $\mathbb{Q}$ of the $2n$-tuple $({\lambda}_{1},\cdots ,{\lambda}_{n},{e}^{{\lambda}_{1}},\cdots ,{e}^{{\lambda}_{n}})$ is at least $n$ for all ${\lambda}_{1},\cdots ,{\lambda}_{n}\in \u2102$ 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 $\u2102$ 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.

Revised:

Accepted:

Published online:

Keywords: Schanuel’s conjecture, forcing and generic absoluteness

Matteo Viale ^{1}

@article{CML_2016__8_2_59_0, author = {Matteo Viale}, title = {Forcing the truth of a weak form of {Schanuel{\textquoteright}s} conjecture}, journal = {Confluentes Mathematici}, pages = {59--83}, publisher = {Institut Camille Jordan}, volume = {8}, number = {2}, year = {2016}, doi = {10.5802/cml.33}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.33/} }

TY - JOUR AU - Matteo Viale TI - Forcing the truth of a weak form of Schanuel’s conjecture JO - Confluentes Mathematici PY - 2016 SP - 59 EP - 83 VL - 8 IS - 2 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.33/ DO - 10.5802/cml.33 LA - en ID - CML_2016__8_2_59_0 ER -

Matteo Viale. Forcing the truth of a weak form of Schanuel’s conjecture. Confluentes Mathematici, Volume 8 (2016) no. 2, pp. 59-83. doi : 10.5802/cml.33. https://cml.centre-mersenne.org/articles/10.5802/cml.33/

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