Non-archimedean analysis on the extended hyperreal line ${}^{\ast }{R}_d$ and the solution of some very old transcendence conjectures over the field $Q$

Abstract

In this paper possible completion ${}^{\ast }{R}_d$ of the Robinson non-archimedean field ${}^{\ast }R$ constructed by Dedekind sections. Given an class of analytic functions of one complex variable $f\in C[z]$,we investigate the arithmetic nature of the values of $f$ at transcendental points $e^n$. Main results are: 1) the both numbers $e+\pi$ and $e\pi$ are irrational, 2) number $e^e$ is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtainedNon conservative extensions of the canonical nonstandard analysis also is considered.

139pp.1figures arXiv admin note: text overlap with arXiv:math/0604466 by other authors

References in corpus (2)

• Non-archimedean analysis on the extended hyperreal line ${}^{\ast }{R}_d$ and the solution of some very old transcendence conjectures over the field $Q$
• A complete axiomatization of infinitary first-order intuitionistic logic over ${\mathcal{L}}_{{\kappa }^{+},\kappa }$

Cited by in corpus (2)

• There is No Standard Model of ZFC and ZFC_2 with Henkin semantics.Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms.Consistency Results in Topology
• Non-archimedean analysis on the extended hyperreal line ${}^{\ast }{R}_d$ and the solution of some very old transcendence conjectures over the field $Q$