Assuming the fact that the set of algebraic numbers is countable, prove that the set of transcendental numbers has the same cardinality R, the set of real numbers.
The Attempt at a Solution
Let A be the set of algebraic numbers and T the set of transcendentals. Then R= A U T, so if T was countable then so would R be (because a countable union of countable sets is countable). Contradiction. Thus T is uncountable.
But this only proves that T is uncountable, and we need to prove MORE than that, namely, |T|=|R|. How to prove this?
Any help is appreciated!