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Comment by baddox | Hacker News Reader

Let g = Graham's number. Then use Knuth's up arrow notation. Then (and this is perhaps cheating) use a non-computable function like the busy beaver function, which grows faster than ANY computable function. That last step may in fact not be cheating, because it might be possible to prove a lower bound and therefore prove that it's higher than the opponent's number.

Pretty notation here:

http://mathbin.net/50035

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This is what I was thinking. Knuth's up arrow on Graham's number. I like the addition of the busy beaver function. A creative improvement for sure.

So far I think you win :)

for the lazy amongst us: http://en.wikipedia.org/wiki/Knuths_up-arrow_notation

If you use Busy Beavers, shouldn't you go all out and do something like that:

BB(BB(BB(BB(BB(…g↑↑↑↑g)))))

or a mix thereof?

edit: I'd actually try BB(BB(BB(BB(BB(…g↑↑↑↑g)))))+1, just in case.

There's no end to that. You could define a function BZ(n) that equals BB(BB(BB...(g↑↑↑...g)...)) with n nested BB's and n up arrows. Then you could define a function RBB(n) that equals BZ(BZ(BZ...(g↑↑↑...g)...)) with n nested BZ's and n up arrows.

If the busy beaver function runs on a computer, it is computable, unless I missed something pretty interesting.

A busy beaver is a Turing machine that runs for more steps than any other machine (having the same state machine size) but eventually halts (rather than looping forever). The busy beaver function measures the run times of all busy beavers, the maximum number of steps you can get out of each state machine size. Apparently Radó proved this value eventually grows larger than any computable function.

http://en.wikipedia.org/wiki/Busy_beaver#Non-computability_o...

The busy beaver function doesn't run on a computer. We only know Σ(n) for n < 5. I think you are missing out on something extremely interesting—that there are limits on what computers can compute, even given unbounded resources and time.

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