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Comment by Jaxan | Hacker News Reader

I don’t think such a bound in itself is interesting, unless the proof reveals some strategy to prove more. There are infinitely potential bounds to BB(n), why is this particular case BB(10) > G useful for understanding fast growing functions?

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It's impressive that a Turing machine of only 10 states can run for Graham's number steps and still halt.

I'd call it absolutely mind boggling, actually.

> why is this particular case BB(10) < G useful for understanding fast growing functions?

it's... not?

it is useful for understanding BB()

First, an upper bound would be more useful to understand BB. But second, out of all large numbers, does this one give any information? We’ll have to see. The fact in itself is useless.

> First, an upper bound would be more useful to understand BB.

you can't have upper bound without breaking Turing and solving halting problem

> But second, out of all large numbers, does this one give any information? The fact in itself is useless.

yes, it gives info of "OMG BIG NUMBER I KNOW ABOUT, THAT'S SO COOL" quality, like all the rest of recreational mathematics :)

Graham's number is simply a nice checkpoint that invites everyone to learn more about function hierarchies

> you can't have upper bound without breaking Turing and solving halting problem

Not in general, no. But for a fixed n, you can. For example, people have settled the exact value of BB(5).

You reach the exact value by tightening the bound over and over, until one value remains.

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