# 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 TheSetOfAllRealTranscendentalNumbersIsUncountable 2013-03-22 16:08:05 2013-03-22 16:08:05 gilbert_51126 (14238) gilbert_51126 (14238) 11 gilbert_51126 (14238) Theorem msc 03E10