[pash]

As Alonzo Church and Alan Turing showed, the computable numbers [0] are countable too. The computable numbers include all the algebraic numbers and some transcendental numbers (including π and e), so the reals are uncountable "because" of those other transcendentals, the uncomputable numbers. Put differently, almost all reals are uncomputable.

0. https://en.wikipedia.org/wiki/Computable_number

[NAFV_P]

Like Omega:

http://en.wikipedia.org/wiki/Chaitin's_constant

[GregBuchholz]

Can't mention Omega without linking to:

Meta Math!

http://arxiv.org/abs/math/0404335

[NAFV_P]

Good book, I read it a few years ago.

His own proof that there are infinitely many primes is a good head spinner.

[Ind007]

Such a good read.Thanks for sharing this.

[JonnieCache]

Check him out on youtube too, he's a fun speaker.

[jsbgir]

This is really interesting. We could take it further and say that, given that some uncomputable reals have a finite definition (e.g. "the probability that a random algorithm halts"), there is a countable number of definable reals (by assigning a Godel number to each definition), so the uncountability of the reals is strictly due to indefinable numbers!

[surement]

I've only ever heard of one uncomputable number (Chaitin's constant), so this is rather mind-blowing.