[chrismonsanto]

Your insistence here & in the other sub-thread that these numbers are represented as a string of digits suggests that you don't understand the argument being made.

If you don't insist on this, then let me give a quick counterexample to "All the numbers larger than say... a googleplex": googleplex + 1

[dragontamer]

Sure. But my own understanding of your stance has evolved throughout today. My current counter argument is:

> Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely

I know you cannot do this, and I presume you also know you cannot do this. Just because there exists a finite representation doesn't mean you can tell me that representation.

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I insist upon this because your requirement of "finite representation" is seemingly arbitrary to me. We can enumerate all important numbers as say... all numbers represented by algebra (addition, subtraction, multiplication, division, roots, exponents, variables, logs, sine, cosine, integral, derivatives, Knuth's notation, and other functions... etc. etc.) that can be described in fewer than 1-million symbols.

And now we have a significant number of "irrational" and "i" numbers, specifically the set that we'll figure out with modern mathematics. Its a finite set (by bounding it by 1-million symbols, we have a finite number of numbers), and arguably the more important set of numbers that represents how modern math functions.

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I dare say that all "important" numbers follows my (relatively arbitrary) definition above (all numbers describable in 1-million modern mathematical symbols or fewer), and is certainly more important than say... most of the numbers between googleplex and Graham's number.

[chrismonsanto]

>your stance

Just to be clear, I'm not the person you were originally replying to. You can describe what you're talking about by writing a program--whether or not it terminates in our lifetime does not change that it is in fact a representation of the number.

[dragontamer]

I can confidently say that the space to uniquely describe a truly uniformly random number between googleplex and graham's number is so large, THE PROGRAM cannot be written down in this universe even with all the atoms in the universe at our disposal.

Graham's number is very very very large. It is finite, it is an integer, but it is absurdly huge. Graham can describe Graham's number, but arbitrarily / uniformly picking a number close to it at random is basically impossible.

[chrismonsanto]

Graham's number can be computed in a few lines of code using Knuth's up-arrow notation.

[dragontamer]

> 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.

[chrismonsanto]

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.

[1]: https://codegolf.stackexchange.com/questions/83873/theoretic...