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As Alonzo Church and Alan Turing showed, the computable numbers [0] are countable too. The computable numbers include all the algebraic numbers and some transce
by pash 12y ago
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 https://en.wikipedia.org/wiki/Computable_number
- NAFV_P 12y agoLike Omega: http://en.wikipedia.org/wiki/Chaitin's_constant http://en.wikipedia.org/wiki/Chaitin's_constant
- GregBuchholz 12y agoCan't mention Omega without linking to: Meta Math! http://arxiv.org/abs/math/0404335 http://arxiv.org/abs/math/0404335
- NAFV_P 12y agoGood book, I read it a few years ago. His own proof that there are infinitely many primes is a good head spinner.
- Ind007 12y agoSuch a good read.Thanks for sharing this.
- JonnieCache 12y agoCheck him out on youtube too, he's a fun speaker.
- surement 12y agoI've only ever heard of one uncomputable number (Chaitin's constant), so this is rather mind-blowing.
- jsbgir 12y agoThis 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!