Transcendental Numbers are Uncountable

Theorem
The set of transcendental real numbers is uncountable.

Proof
By definition, a transcendental number (in this context) is a real number which is not an algebraic number.

Recall that the Real Numbers are Uncountable.

Also recall that the Algebraic Numbers are Countable.

The result follows from Uncountable Set less Countable Set is Uncountable.