Thread: Prove the existence of uncountably many transcendental numbers

It says we need to use the theorem
"The union of a finite or countable number of countable sets A1,A2,... is itself countable." &
"The set of real numbers in the closed unit interval [0,1] is uncountable."

Here is a hint. All algebraic numbers are roots of non-zero polynomials with rational coefficient. There are countable many non-zero polynomials with rational coefficients and each of these polynomials has finitely many zeros. This means that the cardinality of the algebraic numbers is countable. Can you see what to do next?

The set of real numbers is a union of Algebraic numbers and transcendental numbers and uncountable. The set of Algebraic numbers is countable. Therefore the set of transcendental numbers should be uncountable, otherwise the set of real numbers would be countable.

Originally Posted by ninano1205

The set of real numbers is a union of Algebraic numbers and transcendental numbers and uncountable. The set of Algebraic numbers is countable. Therefore the set of transcendental numbers should be uncountable, otherwise the set of real numbers would be countable.

Exactly.