M2C2A: Episode 8, The magnitude of infinity.

The best way to see the magnitude of infinity, is to compare it with other big numbers and say "Infinity is bigger than that.". But with what big numbers?
In third place in the list of very big numbers, we have the googol (not to be confused with the Google searching machine), which is equivalent to 10¹⁰⁰ (A.K.A 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000), which is quite a big number.
In second place, we have the googolplex, equal to 10^googol (A.K.A "A number so big it couldn't be written because the observable universe isn't big enough").
But in first place, we have Graham's number, which is equal to this:
Suppose you want to write 3 cubed. You could say 3³, but you could also say 3↑3. But what if you wrote 3↑↑3? That would be 3 to the power of  3↑3. And 3↑↑↑3? This would be 3 to the power of 3↑↑3. But this is already 3^{7,625,597,484,987}! Why would someone need a number so ridiculously big?! Is this Graham's number?
Not even close.


We need 3↑↑↑↑3! And this isn't still Graham's number!!!
Now, let's call 3↑↑↑↑3 g_1. g_{1 is EXTREMELY BIG, but it is NOT Graham's number.}
Now, let's make g₂, where g₂ is equal to 3↑...↑3, and the number of arrows is g₁. Is this Graham's number?
Not by a googol of zeroes close (literally)!!!


Graham's number isn't achieved until g₆₄, and is so ridiculously large, that NOBODY KNOWS HOW MANY DIGITS IT HAS NOR IN WHICH DIGIT DOES IT START.
But this has nothing to do with infinity, does it? But this is the point I wanted to talk about:
Next time you hear "Infinity", remember this:
"BIGGER THAN GRAHAM'S NUMBER".

-The Roaring Thunder