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Matteo Viale
Forcing the truth of a weak form of Schanuel’s conjecture
Confluentes Mathematici, 8 no. 2 (2016), p. 59-83, doi: 10.5802/cml.33
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Class. Math.: 03E57, 03C60, 11U99
Mots clés: Schanuel’s conjecture, forcing and generic absoluteness

Résumé - Abstract

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.

Bibliographie

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[10] Andrea Vaccaro & Matteo Viale, “Generic absoluteness and boolean names for elements of a Polish space”, (2016), To appear in Bollettino Unione Matematica Italiana
[11] A. J. Wilkie, Some local definability theory for holomorphic functions, Model theory with applications to algebra and analysis. Vol. 1, London Math. Soc. Lecture Note Ser. 349, Cambridge Univ. Press, Cambridge, 2008, p. 197–213 Article
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