Graham's Number
myscienceacademy.org
myscienceacademy.org
The record for that probably is Goodstein's function. See http://en.wikipedia.org/wiki/Goodstein%27s_function#Sequence... for more. At 12 it already has surpassed Graham's number.
The result that this function was used to prove is that there exist computable functions that cannot be proven to be computable using the Peano axioms. And the reason that it cannot be proven computable is that it grows faster than any function that can be proven computable using the Peano axioms. (Ackermann's function is an example of a fast-growing function that can be proven computable using the Peano axioms, and therefore which must grow more slowly than Goodstein's function.)
Basically anything that you can prove using direct calculation and induction a finite number of times on the integers can be proven starting from the Peano axioms. It is not obvious that there are computable functions that those methods of proof do not suffice for. Goodstein's function grows too fast for this to suffice for.
The proof that it is actually computable requires transfinite induction on the countable ordinals.
This is really cool and interesting, but I have to also admit to being deeply skeptical. I'm going to try to read a lot more on the proof, but it seems incredible (in a negative sense) that transfinite induction could prove that a finite number of operations suffices for anything.
Similarly, "computable" and "transfinite" would seem on their face to never belong in the same proof.
Obviously I don't have the math background to make any claims, but right now I can't convince myself that this could possibly prove what people claim it does. (Edit: Or maybe a better way to state my objection is, it seems to me like the axioms one would need to accept to prove this might not really make sense when trying to describe computable systems.)
First let's take a slightly more verbose notation than the wiki link uses. Let's say that B(b, b', n) is the operation put n into hereditary base b notation, then change the base to b'. Let's define G(b, n) as 1 if n = 0, and otherwise as 1 + G(b+1, B(b, b+1, n) - 1). Then the original G(n) is just G(2, n).
If n < b, then by induction on n we can easily prove that G(b, n) is well-defined (here being well-defined is the same as being a finite number), in fact it is just n + 1.
If n = n_1 b+n_0, with n_1 and n_2 < 0, then by induction on n_1 we can easily prove that G(b, n) is well-defined.
Now by induction we can easily prove that if 2, n_0, n_1, n_2 are < b then G(b, n_0 + n_1 b + n_2 b^2) is well defined.
We can go farther and by induction we can prove that if m, n_0, n_1, ..., n_m < b then G(b, n_0 + n_1 b + n_2 b^2 + ... + n_m b^m) is also well-defined.
Now we're in a position to prove that G(b, b^b) is well-defined. After that we repeat the previous proof and find out that if k < b^b then G(b, k + b^b) is well-defined.
Now we can do an induction proof that G(b, 2 b^b) is well-defined.
And so on. In fact what we discover if we're careful is this. For each hereditary base pattern, there exists an inductive proof that G(b, n) with that hereditary base pattern is well-defined. The challenge is that the necessary proof changes as the hereditary base patterns get bigger.
What transfinite induction provides is a way to describe, in a unified way, the underlying idea behind all of those different inductive proofs into a single one. In particular every hereditary base pattern is mapped onto a (possibly infinite) ordinal. The operation B(b, b', n) does not change which ordinal you're mapped on to. But the operation -1 always decreases the ordinal. Thus we get a decreasing sequence of ordinals. And transfinite induction says that all decreasing sequences of ordinals must terminate in a finite number of steps. The pattern of the necessary inductive proofs is dependent upon the ordinal.
What the Peano axioms lack is any way to unify all of those individual inductive proofs into a single one. Thus for any given integer n we can come up with a proof that G(b, n) is well-defined. But we can't prove that fact for all positive integers n with a single proof.
Therefore this is really an exercise about the limitations of the set of possible proof techniques that the Peano axioms provide, and not about the limitations of computability.
(Of course the limitations of computability on real machines are much, much stricter than the limitations of the Peano axioms. We can't say much that is useful about G(12) other than, "Unbelievably big."
He's done his bit on Graham's number in a couple of them before.
Details of the FotSN tour can be found here:
[0] http://en.wikipedia.org/wiki/Graham%27s_number
[1] http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
You might think that I'm saying that in jest, but it is actually the way of things. There are quite literally an infinite number of numbers. Why should Graham's be important enough for us to consider. For that matter, why should we consider 1?
The answer lies in part in the question - why - it must be important in some way. That is exactly the case - we only consider numbers that, for some reason, relate to problems we are facing, whether it be in mathematics, programming, or grocery shopping, and even then not on their own merits. Past a certain point, you don't hold numbers in your head as little balls, but rather their decimal encoding. That's a clever hack, but not without its downsides. Ask some people randomly if they were in a position to decide, how much budget would they allocate to save 10000 birds, and ask others the same with 100000 birds. The mean of the second answer would not be ten times that of the first (unless your subjects know and actively try to work around that particular bias).
But back to Graham's number - if you think the only way to imagine it is to hold its decimal notation in your head - the obvious question is why? Why not hexadecimal? Octal? Why any base-p encoding at all? The usual reason to use base-10 is to quickly get an idea of the magnitude of a number relative to things we are familiar with and do some simple arithmetical manipulations on it. Both of these are pointless for Graham's number. Relative to familiar things like billions and trillions it's quite a ways of the chart, and adding, subtracting, or even raising to powers of such makes relatively no difference.
Besides which, in this case it isn't really pointless, it's impossible. Jokes about collapsing into black holes aside, there aren't enough bits of information in the universe to encode the decimal representation of Graham's number. There aren't enough bits of information to encode the number of digits of that. Not enough even to encode the number of digits of the number of digits. Interestingly, there also aren't enough bits to encode the number of times you'll have to repeat taking the base-10 logarithm to get back to a universal scale.
Which goes to show that not only your mind, but no entity in the universe can accomplish the feat of representing an arbitrary number in decimal notation, mathematician or otherwise. And what is the point, really? G is a perfectly good symbol, and so is g_64. Usefulness is important, not imagining a string of balls. For that matter, I bet you've been perfectly content to use Pi on at least a few occasions, even though in terms of representability in decimal, it is infinitely worse than Graham's number.
I'm not sure how good an example this is to argue your point, because the second answer should only be ten times the first if the cost structure is a linear function with a zero constant term. Which seems like an awfully big assumption -- you might get economies of scale (nonlinear, decreasing derivative), or you might pick the low-hanging fruit first and then have to go after more difficult birds (nonlinear, increasing derivative), or you might have fixed costs that give you a constant term.
That being said, many people do have a lot of difficulty conceptualizing the difference between millions, billions and trillions [1].
Nevertheless - very interesting topic! Also terrifying.