the set of all real transcendental numbers is uncountable

Theorem.The set of all real transcendental numbers is uncountable.

Proof. Denote $\mathbb{T}$ and $\mathbb{A}$ be the set of real transcendental and real algebraic numbers respectively. Suppose $\mathbb{T}$ is countable. Then the union $\mathbb{T}\cup\mathbb{A}=\mathbb{R}$ is also countable, since $\mathbb{A}$ is also countable, which is a contradiction. Therefore $\mathbb{T}$ must be uncountable. $\Box$

Title	the set of all real transcendental numbers is uncountable
Canonical name	TheSetOfAllRealTranscendentalNumbersIsUncountable
Date of creation	2013-03-22 16:08:05
Last modified on	2013-03-22 16:08:05
Owner	gilbert_51126 (14238)
Last modified by	gilbert_51126 (14238)
Numerical id	11
Author	gilbert_51126 (14238)
Entry type	Theorem
Classification	msc 03E10