[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.