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

^{1}

@article{CML_2016__8_2_59_0, author = {Viale, Matteo}, 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 = {http://www.numdam.org/articles/10.5802/cml.33/} }

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. http://www.numdam.org/articles/10.5802/cml.33/

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

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

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

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

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

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

[8] A classification of type $\mathrm{I}$ $A{W}^{*}$-algebras and Boolean valued analysis, J. Math. Soc. Japan, Volume 36 (1984) no. 4, pp. 589-608 | DOI

[9] $\text{C}{\phantom{\rule{0.166667em}{0ex}}}^{*}$-algebras and $\mathsf{B}$-names for complex numbers, University of Pisa, September (2015) (Thesis for the master degree in mathematics)

[10] Generic absoluteness and boolean names for elements of a Polish space (2016) (To appear in Bollettino Unione Matematica Italiana)

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

*Cited by Sources: *