G(64)
Graham's Number
Graham's Number, sometimes abbreviated as G, was first coined in 1977 when Ronald Graham used it as an upper bound to a problem in Ramsey theory. At the time it was the largest number anyone had seen arising from a serious mathematical proof. It quickly gained recognition in the Guinness Book of World Records for being the largest number in serious mathematics. But what is Graham's Number, and how is it constructed?
It can be defined quite simply by using Donald's Knuths up-arrow notation and a new recursive function, we will call the G-function. The G-function is defined as follows:
Let G(n) be defined when "n" = {1,2,3,...}:
G(1) = 3^^^^3
if n>1 then G(n) = 3^^^^ ... ... ^^^^3 w/G(n-1) ^s
Graham's Number can then be defined as G(64).
The number is notable for being the largest number most people know about, or have heard of. In short, it encapsulates the popular notion of "the largest number ever". This is a notion that professional mathematicians are quick to dismiss since there is "no largest number", but that has not stopped lay people from having a general sense that such a thing should exist. After all, human history is finite, so there must be some largest well defined finite number at any given point in time. The problem usually is that it's difficult to say whether a number (when sufficiently large) is well-defined or not. In the case of Graham's Number however, the number is "so small" that it's well-foundedness is quite clear. If every expression in Up-arrow notation is valid, then Graham's Number must also be valid. Since each new up-arrows validity is built upon the validity of the previous up-arrow the entire system rests upon the validity of exponents. Essentially Graham's Number is nothing more than a REALLY large power of 3. There is no doubt then that Graham's Number, despite its size, is a meaningful and well-defined number.
We are generally lead to believe that Ronald Graham did not make Graham's Number solely for the purpose of being large, and this is what grants it "validity". While I can't really comment upon the proof itself, I can say that there clearly is a relation between the numbers size and why a paper about it got published. The original paper was not an official paper, but it got attention from another mathematician specifically for the upper-bounds size. So even from the beginning the size of Graham's Number was more than just a side note to the proof. It was the proofs claim to fame! It may not have garnered as much interest otherwise. This begs the question: are professional mathematicians any less enamored of large numbers than so called "googologists"? I think the fascination of large numbers is nigh universal and professional mathematicians disguise their own predilection for them, from time to time, with a certain claim to "serious mathematics". It should be noted that the problem from which Graham's Number arises from is probably no more practical, from the laymen's point of view, than the number itself. Of coarse the biggest blow to any claim Graham's Number has to "serious mathematics" is that the actual solution to the problem is mostly likely absurdly smaller, making the "upper-bound" practically useless. In any case, Graham's Number, despite its popular status, is no longer the largest number used in a serious mathematical proof, as others have come after to claim that title.
For our purposes of coarse, Graham's Number is nothing particularly special outside of its historical significance. It in fact has one foot in a familiar landscape of up-arrows, and it gazes out on a vast field of glorious transcendence! In a very real sense Graham's Number really just takes the first step towards something really amazing, then inexplicably stops in its tracks! Still Graham's Number serves as a good bench mark for large numbers. If you're number doesn't even make it this far, then you're really not a contender.
People new to large numbers often like to hijack this number to make some "truly enormous numbers" beyond Graham's Number. Obviously we can have G+1, or 2G, or G^2. These aren't very clever however since they are simple arithmetic extensions that anyone could come up with. Slightly better would be G^G^G^...^G^G w/G Gs. However this is merely G^^G. Why not G^^^G, or G^^^^G. Yet all of this is actually a very small improvement! The best way to proceed where Graham left off is to use a larger input for his own G-function. As you will see, I will have a number of these expressions listed here, but they can all easily be captured and overtaken by more powerful and general methods! We've only really begun with large numbers! As Ronald Graham says himself: "Graham's Number is really no closer to infinity than the number 1 is. Even though it took so many steps to get to Graham's Number, it takes many more, infinitely more, to get to infinity!". I couldn't say it better myself, and since we've got an infinite number of steps left to go we'd better get started ...