Graham's Number

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How many knew about this? It's the largest number ever used in a mathematical proof. It is impossible to express in scientific notation, or even as a tower of exponents such as 3^(3^(3^(3^..... Here is my best explanation of it:

Just as exponentiation (^) is iterated multiplication, let the ^^ operator be iterated exponentiation. Then let ^^^ be iterated ^^, ^^^^ be iterated ^^^, and so on. A few examples:

3^3 = 3*(3*3) = 27.
3^^3 = 3^(3^3) = 3^27 = 7,625,597,484,987.
3^^^3 = 3^^(3^^3) = 3^^7,625,597,484,987 = a tower of exponent 3's that is 7,625,597,484,987 layers high. Already far greater than a googolplex (10^(10^100))! !!

Let G(1) be 3^^^^3.
Think that's huge? Let G(2) be 3^^^...(with 3^^^^3 carats)...^^^3.
Let G(n) be 3^^^...(with G(n-1) carats)...^^^3.

Graham's number is G(64).

wiki to the rescue
http://en.wikipedia.org/wiki/Kruskal%27 ... inite_form

When you get a chance look up the Ackerman Function.

See: http://en.wikipedia.org/wiki/Ackerman_function

ruveyn

I beleive the googolplex is bigger.

Yes, TREE(3) is bigger than Graham's number. The stipulation I put in my original post, however, was "used in a mathematical proof." You can have TREE(4) or TREE(10) or TREE(Graham's number) or A(TREE(Graham's number)), but have they ever been put to use in a mathematical proof?

Maybe TREE(3) has been used in a mathematical proof now? My stipulation may have been outdated.

Graham's number, however, is still far, far, far greater than a googolplex. As I stated earlier, 3^^^3 comes to a tower of exponent 3's that is 7,625,597,484,987 layers high. And Graham's number uses an operator with unimaginably more carats than that. A googolplex, on the other hand, is only 10^(10^100)!

10^(10^100)
< 27^(27^243)
= 27^((3^3)^243)
= 27^(3^(243*3))
= 27^(3^729)
= (3^3)^(3^729)
= 3^((3^729)*3)
= 3^(3^730)
<< 3^(3^7,625,597,484,987)
= 3^(3^(3^(3^3)))
<<< a tower of exponent 3's that is 7,625,597,484,987 layers high
<<<< Graham's Number.

There is absolutely no limit to how large a number can get before becoming infinity. Something far more ridiculous than A(TREE(Graham's number)), such as TREE(TREE(TREE(TREE(...with TREE(1,000,000) layers...(TREE(TREE(1,000,000)))))...)))), would STILL be less than infinity.

I do like the idea that we know that "The last ten digits of Graham's number are ...2464195387." In fact, we know a few more.

It would appear from the article that

Indeed, after looking around a bit more, I've found that the proof with Graham's Number in it was written BEFORE I WAS BORN. Yes, there are larger numbers now that have been used in mathematical proofs.

'Infinity' is not simply a very huge number, although that is part of it.. infinity can be anything, or nothing.

Yeah, in calculus they use infinity in various ways to get results that seem to match reality (or vice versa), but literally speaking, infinity is anything.. although since it's infinity, it isn't entirely literal, and since words are literal (by definition ) this whole post is infinitely inadequate, but that's why it's infinity

G(65); there, my number is bigger.

Great, now we're getting into degrees of infinity...

Let N be the set of natural numbers. That is an infinite set. Graham's number and TREE(3) are both in this set.

Let F be some finite set larger than zero elements.

P(F) is clearly a larger set than F.

So it kinda strikes me that P(N) should be larger than N, even though both are infinite sets! If that is the case, then by mathematical induction P(P(P(P(P(...with Graham's number of layers...(P(P(P(N))))...))))) is still larger.