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.

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

1 Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
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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/

[1] James Ax On Schanuel’s conjectures, Ann. of Math. (2), Volume 93 (1971), pp. 252-268

[2] Thomas Jech Set theory, Spring Monographs in Mathematics, Springer, 2003 (3rd edition)

[3] Thomas J. Jech Abstract theory of abelian operator algebras: an application of forcing, Trans. Amer. Math. Soc., Volume 289 (1985) no. 1, pp. 133-162 | DOI

[4] Jonathan Kirby Exponential algebraicity in exponential fields, Bull. Lond. Math. Soc., Volume 42 (2010) no. 5, pp. 879-890 | DOI

[5] Jonathan Kirby; Boris Zilber Exponentially closed fields and the conjecture on intersections with tori, Ann. Pure Appl. Logic, Volume 165 (2014) no. 11, pp. 1680-1706 | DOI

[6] Kenneth Kunen Set theory, Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing Co., Amsterdam-New York, 1980, xvi+313 pages (An introduction to independence proofs)

[7] David Mumford Algebraic geometry. I, Classics in Mathematics, Springer-Verlag, Berlin, 1995, x+186 pages (Complex projective varieties, Reprint of the 1976 edition)

[8] Masanao Ozawa A classification of type I AW * -algebras and Boolean valued analysis, J. Math. Soc. Japan, Volume 36 (1984) no. 4, pp. 589-608 | DOI

[9] Andrea Vaccaro C * -algebras and B-names for complex numbers, University of Pisa, September (2015) (Thesis for the master degree in mathematics)

[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.), Volume 349, Cambridge Univ. Press, Cambridge, 2008, pp. 197-213 | DOI

[12] B. Zilber Pseudo-exponentiation on algebraically closed fields of characteristic zero, Ann. Pure Appl. Logic, Volume 132 (2005) no. 1, pp. 67-95 | DOI

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