TITLE: Is there a way of intuitively grasping the magnitude of Graham's number?
QUESTION [15 upvotes]: I have heard it stated before that Graham's number is so vast that it is completely beyond comprehension. It is way larger than the number of atoms in the universe, so cannot be related to real quantities, and unfathomably larger than even hypothetical quantities, such as the number of Planck unit-sided cubes that could fit in the observable universe.
People usually stop at this point, and state that it is simply beyond comprehension, which is fair enough. However, I want to know more about what numbers are within comprehension, and what "intutitional" devices can be used to comprehend them.  For instance, I considered combinatorial analogies, such as the number of different ways there are of selecting half the atoms in the universe (which contains $10^{80}$ atoms):
$$10^{80} \choose 5 \times 10^{79}$$
or the number of different ways there are of visiting every single atom in the universe exactly once ("universal atom tours"?):
$$10^{80}! \over 2$$
or even, the number of ways of selecting half the universal atom tours!
$${10^{80}! \over 2} \choose {10^{80}! \over 4}$$
Clearly we can continue in this manner, sacrificing semantic succinctness and intuitive meaning for further gains in magnitude. Even if, as I imagine is the case, these quantities barely scratch the surface of Graham's number, I suppose I am asking two things:

How can the magnitudes of extremely large constructs like the ones I've given be evaluated and compared?
Even if Graham's number is out of reach, what are some particularly effective 'intuitional devices' for explaining the scale of extremely large quantities to laymen like myself?

Thanks.

REPLY [2 votes]: I'm going to take this one step further than quantus14.  Once you reach Graham's number, the question then becomes

How to effectively compare and understand even larger numbers?

Well, the first thought should be to see if you can find a similar number to yours on the Googology Wiki, which is dedicated entirely to large numbers and their study.
My second recommendation is to learn fast growing hierarchy (FGH), which has a similar construction to Knuth's up-arrow notation, but it extends much further.  Here is a good YouTube series.
We define the first two rules of FGH as follows:

$f_0(n)=n+1$
$f_{\alpha+1}(n)=\underbrace{f_\alpha(f_\alpha(\dots f_\alpha(n)\dots))}_{n\text{ amount of }f_\alpha's}$

A quick demonstration will show you how fast this grows:
$$\begin{align}f_1(5)&=f_0(f_0(f_0(f_0(f_0(5)))))\\&=f_0(f_0(f_0(f_0(6))))\\&=f_0(f_0(f_0(7)))\\&=f_0(f_0(8))\\&=f_0(9)\\&=10\end{align}$$
$$\begin{align}f_2(5)&=f_1(f_1(f_1(f_1(f_1(5)))))\\&=f_1(f_1(f_1(f_1(10))))\\&=f_1(f_1(f_1(20)))\\&=f_1(f_1(40))\\&=f_1(80)\\&=160\end{align}$$
$$\begin{align}f_2(160)&=\underbrace{f_1(f_1(\dots f_1(160)\dots))}_{160}\\&=5\times2^{165}\\f_3(5)&=f_2(f_2(f_2(f_2(f_2(5))))\\&=f_2(f_2(f_2(f_2(160)))\\&=f_2(f_2(f_2(f_2(f_2(5))))\\&=f_2(f_2(f_2(5\times2^{165}))\\&=f_2(f_2(\underbrace{f_1(f_1(\dots f_1(5\times2^{165})\dots))}_{5\times2^{165}})\end{align}$$
Indeed, you will find these numbers quickly exceed anything you could hope to write on a sheet of paper.
We also have approximations to your numbers, and by a quick check, they are around the range of $f_4(n)$ for some small values of $n$.
...and then we have rule 3:

$f_\alpha(n)=f_{\alpha[n]}(n)$

This rule is a little bit confusing, so let me explain:
$\alpha$ is a limit ordinal, which basically translates into the limit of some sequence of numbers (a list of numbers (or a list of lists) if you are a programmer):
$$\alpha=\sup\{\alpha[1],\alpha[2],\alpha[3],\dots\}$$
We first define $\omega$ to be the limit of the following sequence:
$$\omega=\sup\{1,2,3,\dots\}$$
We then have $\omega[5]=5$ as the fifth term of this sequence, so
$$f_\omega(5)=f_{\omega[5]}(5)=f_5(5)$$
And from there we know this is very large...much larger than the previous numbers.
To deal with $\omega+1$, we apply the second rule, followed by the third:
$$\begin{align}f_{\omega+1}(5)&=f_\omega(f_\omega(f_\omega(f_\omega(f_\omega(5)))))\\&=f_\omega(f_\omega(f_\omega(f_\omega(f_5(5)))))\end{align}$$
So you can see how large this gets, and it gets pretty big.  Graham 's number has a nice approximation:
$$f_{\omega+1}(64)\approx\text{Graham 's number}$$
So this is rather simple and
$$f_{\omega+1}(64)=\underbrace{f_\omega(\dots f_\omega(64)\dots)}_{64}$$
Note that we can even go further:
$$\omega\cdot2=\sup\{\omega+1,\omega+2,\omega+3,\dots\}$$
For example:
$$\begin{align}f_{\omega\cdot2}(5)&=f_{\omega+5}(5)\\&=f_{\omega+4}(f_{\omega+4}(f_{\omega+4}(f_{\omega+4}(f_{\omega+4}(5)))))\end{align}$$
And we would then expand further and further and further...
And we have some other larger ordinals:
$$\omega^2=\{\omega\cdot1,\omega\cdot2,\omega\cdot3,\dots\}$$
And beyond!
$$\omega^\omega=\{\omega^1,\omega^2,\omega^3,\dots\}$$
While it's not very... visual, it does become clear that you can make some very very big numbers while also being able to compare large numbers.  One of the great advantages of FGH is the clearness of how to compare Graham's number to another number.
And the only thing holding you back from producing even larger numbers is how you want to define even larger ordinals.

And one last example:
$$\begin{align}f_{\omega^\omega}(3)&=f_{\omega^3}(3)\\&=f_{\omega^2\cdot3}(3)\\&=f_{\omega^2\cdot2+\omega^2}(3)\\&=f_{\omega^2\cdot2+\omega\cdot3}(3)\\&=f_{\omega^2\cdot2+\omega\cdot2+3}(3)\\&=f_{\omega^2\cdot2+\omega\cdot2+2}(f_{\omega^2\cdot2+\omega\cdot2+2}(f_{\omega^2\cdot2+\omega\cdot2+2}(3)))\\&=\cdots\end{align}$$