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Comment by amelius | Hacker News Reader

Related only to large numbers, but is there some theory about generalizing and extending our usual mathematical operators +, ×, and ^ (power)?

+ applied N times becomes ×N

× applied N times becomes ^N

^ applied N times becomes ...?

etcetera

And would such a theory have any practical use?

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Knuth's up-arrow notation is one of a few ways to address the addition->multiplication->exponentiation->tetration->... extensions: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

As for practicalities, the mathematician in me will let the scientists deal with that.

I like the description of graham's number (by graham) using the up-arrow notation https://youtu.be/GuigptwlVHo?t=31

> And would such a theory have any practical use?

Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function).

[1]: https://en.wikipedia.org/wiki/Ackermann_function

Given that Graham's Number is an (over-)extension of the Ackermann-function that means that it's still computable, correct?

Given the series leading up to Graham's Number, G = g_64 (g_1, g_2, ...), BB(n) will outgrow g_n, right? If so what's the smallest n such that BB(n) > g_n?

After a bit of research I found that there is a proof [0][1] that BB(12) > g_1. But that's still a ways from g_2, let alone g_64.

[0]: "A Lower Bound on Rado's Sigma Function for Binary Turing Machines" by Milton Green (1964)

[1]: https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_....

Graham's number comes from such an extension, and is about 64 such operations deeper. It's useful in the sense that it provides an upper bound for some proof that hasn't yet been proven for ALL numbers.

http://waitbutwhy.com/2014/11/1000000-grahams-number.html

https://en.wikipedia.org/wiki/Hyperoperation

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