Forcing the truth of a weak form of Schanuel’s conjecture
Confluentes Mathematici, Volume 8 (2016) no. 2, pp. 59-83.

Schanuel’s conjecture states that the transcendence degree over of the 2n-tuple (λ 1 ,,λ n ,e λ 1 ,,e λ n ) is at least n for all λ 1 ,,λ n which are linearly independent over ; if true it would settle a great number of elementary open problems in number theory, among which the transcendence of e over π.

Wilkie [11], and Kirby [4, Theorem 1.2] have proved that there exists a smallest countable algebraically and exponentially closed subfield K of such that Schanuel’s conjecture holds relative to K (i.e. modulo the trivial counterexamples, 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.

Published online:
DOI: 10.5802/cml.33
Classification: 03E57, 03C60, 11U99
Keywords: Schanuel’s conjecture, forcing and generic absoluteness
Viale, Matteo 1

1 Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
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Viale, Matteo. 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.

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