How do we know that there are infinitely many transcendental numbers?

Question

How do we know that there are infinitely many transcendental numbers?

In: 4

Answer 1

We can prove that the real numbers are uncountable and the algebraic numbers are countable. If the transcendental numbers were countable, then the union of the transcendentals and the algebraics (i.e. the real numbers) would also be countable. Since the real numbers are uncountable, that assumption must be false, so the transcendentals must be uncountable.