- #1

kingwinner

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## Homework Statement

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.

## Homework Equations

N/A

## 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!