the set of all real transcendental numbers is uncountable

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

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