3 ms·
If all we want to do is prove the existence of transcendental numbers, rather than construct them, that follows as soon as we know: 1. Algebraic numbers are co
by CurtMonash 13y ago
If all we want to do is prove the existence of transcendental numbers, rather than construct them, that follows as soon as we know:
1. Algebraic numbers are countable.
2. Real numbers aren't countable.
#1 is easy -- the number of polynomials with rational coefficients is obviously countable, and they have finitely many real roots each. So pick your favorite proof of #2 and you're done.
- deleted 13y ago[deleted]
- ColinWright 13y agoTrue. And in truth, the linked article is my favorite proof of #2. And that's the point. This proof is really showing the uncountability of the reals - nowhere does it use any property of the algebraics other the fact that they're countable.