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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
Article PDF
Class. Math.: 03E57, 03C60, 11U99
Keywords: 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.


[1] James Ax, “On Schanuel’s conjectures”, Ann. of Math. (2) 93 (1971), p. 252-268
[2] Thomas Jech, Set theory, Spring Monographs in Mathematics, Springer, 2003, 3rd edition  MR 1940513
[3] Thomas J. Jech, “Abstract theory of abelian operator algebras: an application of forcing”, Trans. Amer. Math. Soc. 289 (1985) no. 1, p. 133-162 Article
[4] Jonathan Kirby, “Exponential algebraicity in exponential fields”, Bull. Lond. Math. Soc. 42 (2010) no. 5, p. 879-890 Article
[5] Jonathan Kirby & Boris Zilber, “Exponentially closed fields and the conjecture on intersections with tori”, Ann. Pure Appl. Logic 165 (2014) no. 11, p. 1680-1706 Article
[6] Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics 102, North-Holland Publishing Co., Amsterdam-New York, 1980, An introduction to independence proofs
[7] David Mumford, Algebraic geometry. I, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Complex projective varieties, Reprint of the 1976 edition
[8] Masanao Ozawa, “A classification of type ${\rm I}$ $AW^{\ast } $-algebras and Boolean valued analysis”, J. Math. Soc. Japan 36 (1984) no. 4, p. 589-608 Article
[9] Andrea Vaccaro, $\text{C}\,^*$-algebras and $\mathsf {B}$-names for complex numbers, Thesis for the master degree in mathematics, University of Pisa, 2015
[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
[12] B. Zilber, “Pseudo-exponentiation on algebraically closed fields of characteristic zero”, Ann. Pure Appl. Logic 132 (2005) no. 1, p. 67-95 Article
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