> uniquely describe a truly uniformly random number
This is the hard part. Picking a randomly uniform number "close to" Graham's number.
Graham's number itself is easily described of course: I can just say "Graham's Number". The numbers "close to it", (say, +/- 1% of Grahams number), are impossible to describe.
If you don't believe me, then please ship me the impossible number of hard drives that describes one such number, but you will have had to have picked it truly randomly.
Pick one at random.
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EDIT:
> Graham's number can be computed in a few lines of code using Knuth's up-arrow notation.
Also, this is wrong. The first number of the pyramid can be described in up-arrow notation. But even the 2nd number of the pyramid requires g1 (ie: ~7.6 Trillion) up arrows to describe.
I can safely say that g3 (which requires G2 arrows to describe) has more up-arrows involved than there are atoms in this universe. So g3 already cannot be described by a computer program using up-arrow notation alone. And Graham's number is g64, sitting on top of a huge pyramid of such numbers.
Again, you're implicitly assuming the representation has to be a string of digits. I'm not sure that I can convince you, or even what to convince you of, as you aren't accepting the premises of the argument. The numbers +/- 1% of Graham's number can be trivially represented with a program. Likewise, here's some people codegolfing programs to output Graham's number[1]. If it seems like I'm cheating by using too powerful of a method, that's the point that the original person was making: there exist real numbers that can't be described even like this.
dragontamer|3 years ago
This is the hard part. Picking a randomly uniform number "close to" Graham's number.
Graham's number itself is easily described of course: I can just say "Graham's Number". The numbers "close to it", (say, +/- 1% of Grahams number), are impossible to describe.
If you don't believe me, then please ship me the impossible number of hard drives that describes one such number, but you will have had to have picked it truly randomly. Pick one at random.
-------------
EDIT:
> Graham's number can be computed in a few lines of code using Knuth's up-arrow notation.
Also, this is wrong. The first number of the pyramid can be described in up-arrow notation. But even the 2nd number of the pyramid requires g1 (ie: ~7.6 Trillion) up arrows to describe.
I can safely say that g3 (which requires G2 arrows to describe) has more up-arrows involved than there are atoms in this universe. So g3 already cannot be described by a computer program using up-arrow notation alone. And Graham's number is g64, sitting on top of a huge pyramid of such numbers.
chrismonsanto|3 years ago
[1]: https://codegolf.stackexchange.com/questions/83873/theoretic...