3.2.12 - Greater_than_Graham

Greater than Graham

Introduction

In the last article we explored Graham's Number, a number touted as the "largest number ever used in a serious mathematical proof", despite the fact that this is already 30 years out of date! This hasn't stopped the man in the street however, from considering Graham's Number as the pinnacle of modern mathematical achievements in the field of "coming up with large numbers" (googology). Considering how easy it is to construct Graham's Number with just a handful of recursive definitions, it is not surprising that many a would-be googologist takes it upon themselves to name a value "far greater than Graham's Number". While they generally succeed in this task, generally with relatively little effort, they fail to realize that Graham's Number was never invented to be a difficult number to beat. It was suppose to be the smallest number that could be proven to be larger than the solution to Graham's problem. The fact of the matter is, Graham's problem never required much recursion to begin with! This is why it's so easy to "trounce the grand champion of professional mathematics". It never professed to be any such thing. This is the kinds of numbers mathematicians come up when they're not even TRYING!

When mathematicians do create large numbers, just for theory sake, crazy things like the fast growing hierarchy result. This goes way way way beyond Graham's Number and instead seeks to generalize the concept of recursion to countable hierarchies. If that wasn't bad enough, now days, even the numbers mathematicians actually make use of are ridiculously huge ... much much larger than Graham's Number. A Number called TREE(3) is so huge that there is no way to explain it without explaining a good portion of the fast-growing hierarchy (we won't get to that subject until Section IV). And there is an even larger number called SCG(13) which is arguably the current champion "largest number used in a serious mathematical proof". The only reason Graham's Number has retained the title in the popular imagination is because the category "Largest Number used in a serious mathematical paper" in the Guinness Book of World records was discontinued shortly after 1980. So the general public was simply not made aware of these further developments. It will probably take another Martin Gardner to popularize TREE(3), SCG(13) and the like. Of coarse googologist's have been privy to this for quite a while already because they go out of their way to find these sorts of things out. In truth, it is not difficult for anyone to do a little research and find this out. This just demonstrates that googology is in fact a rather obscure hobby. Most people probably haven't even heard of the humble googolplex. All the more reasons for sites like this one to exist to educate the interested in the art of large numbers.

Prior to the existence of the googology community, the amateur setting his sights on Graham's Number was more excusable. How is the average person to know any better? Yet the googology community is so small, and the interest in large numbers so general and likely to spring up literally anywhere in the world, it is perhaps no surprise that, time and time again, amateurs rise to the challenge to topple the great and terrible Graham's Number. This is often followed by all manner of triumphant boasting.

When I first returned to the study of large numbers, back in 2004, I too fell prey to this tendency, though in my case it was doubly ironic since I had already envisioned numbers much larger than Graham's Number, or any simple extension of it, when I was a kid with my poly-cell hierarchy. It wasn't long however until I discovered chain-arrows and then array-notation, and so this phase was rather short lived.

If your going to come up with a number larger than Graham's would think the next logical thing would be G₆₅. Many people however do all sorts of other things to extend Graham's Number. Perhaps part of the problem is that the growth rate of Graham's function is very counter-intuitive and much faster than we are used to. G₆₅ is much larger than any extension you could create with G₆₄ using elementary operations and Knuth-Arrow Notation, as I'll demonstrate!

In this article we will explore some naive attempts to extend beyond Graham's Number. As we do so, we'll learn the real secret to transcending all of this stuff ... general recursion. I'll even offer some of my own naive attempts at extending Graham's Number back in 2004 when I first learned about the number. Then I'll show how we can use general recursion to totally mop the floor with almost any amateur attempt at extending Graham's Number...

Naive Extensions

What is a naive extension? This is a term I use to express a phenomenon common in googology. Basically a naive extension of any googological system is one which utilizes tools already within that system to "extend" that system. But since those tools are already within the system, it's something the creator of the system would have already anticipated. Therefore it's not worthy of bragging rights. It's the equivalent of taking a work of art, slapping a sticker on it, and claiming you've achieved "more" than the original artist. You wouldn't even have a piece of art to slap a sticker on if they didn't create it first, and it's not been demonstrated that you could create a work of art of equal magnitude. An even worse form of naive extension is any extension of a googological system which utilitizes tools of elementary arithmetic. For example, using addition, multiplication, exponentiation, or factorials. That's the equivalent to a grade schooler scribbling in a calculus book. In terms of googology such elementary operations are negligible in their effect on most of the large numbers studied. Only the very smallest (in the low tetrational range) are actually significantly effected via such methods. In order for an extension to not be naive, it must actually invent something new, demonstrating a full understanding of the original system in order to extend it further. Anything else is simply working within the system, even if, you have technically named an explicit number larger than any explicitly named by the original creator of the system.

Naive extensions are often created in one of two ways. In some situations, a sincere (but naive) attempt is made to come up with a larger number. In these instances the naive extension is simply the result of the persons own inexperience in the subject. Sometimes, this person doesn't even fully understand the notation being used. There is the risk therefore, that they're "extension" might end up being meaningless. On the other hand, if it is meaningful, it probably doesn't improve much on the original number. The second kind of way that naive extensions result is far more insincere. A person will create a naive extension in an attempt to lampoon the whole endeavor and make a mockery of it by saying ... "see even I could come up with a bigger number than that". The result is usually a salad number. A salad number is any haphazard combination of existing notations and googolism's. Both "naive extension" and "salad number" are derogatory terms in googology parlance. However, not all salad numbers are invented maliciously. Sometimes the inventor is simply having a little fun of their own. Sometimes however, it's clear that the naive extension is really just an attempt to dismiss or out do the work of mathematicians and googologists.

This is why I have come to the position that googology is NOT about coming up with the largest number. Googologist's are of coarse well aware that numbers don't end, that you can always add 1! That's the foundation of the subject! But the supposed futility of this doesn't stop googologists, because they are simply enamoured with that endlessness and want to explore it. The goal is not to find the highest number, but to continually find larger and larger ones as long as you can muster. That is a motivation that escapes the nay sayers. For them, it's enough to say "numbers never end", without actually describing any. For the googologist however, this statement is far too vague. What does never end even mean? What kinds of numbers actually exist and what are they like? This is the kind of thinking that drives googology forward, a desire to have an ever more general perspective on what it means to say numbers never end...

In the next sub-heading we'll start with naive extensions of Graham's Number involving only elementary mathematics, then we'll continue to progressively more and more sophisticated attempts to come up with something "greater than Graham".

Elementary Extensions

An elementary extension of a googolism, is the worst kind of extension possible! If you want to embarrass yourself in front of a googologist, just try creating an elementary extension of a googolism and start boosting that "you just created a MUCH larger number". This will result in instant facepalmage from any googologist worth his salt.

Let's define an elementary extension formally. Forget about Graham's Number, or [INSERT GOOGOLISM HERE] for a moment. Whatever it is, however large it is, it's just "N". Say we have a number system which includes decimal notation...

0123456789

... and the following so called elementary functions ...

+ - * / ^ log !

... and the number N.

Any combination of decimal notation, elementary functions, and N is just an elementary extension of N, and for sufficiently large N can be approximated as N. Read that again and think carefully about that ... ANY combination ... you can write ... is an elementary extension, and will seem to be approximately N. In order to escape that using these symbols would require a string with more characters than could possibly fit in the known universe. In fact, that will actually be an understatement ... most of the time!

This is perhaps one of the difficult things for beginners to realize. Our intuitive number sense is so much smaller than the numbers that we deal with in googology that our first inclination is to assume that if you add an exponent here and a factorial there then, by golly, that's got to dwarf anything using that notation. This is not so. If exponentials and factorials were all that were required, then the notations would be superfluous and would not have been invented in the first place! They were invented because the numbers involved COULD NOT be described using exponents and factorials within a reasonable amount of space. Because the numbers are just that big, it also follows that adding in a few extra exponentials and factorials really doesn't mean a whole hell of beans from the googologist's point of view. Remember, in googology the elementary functions aren't worth diddly squat, no matter how large the arguments inside them are. With that in mind let's consider the elementary extensions of Graham's Number.

A common response to Graham's Number, which we will symbolize simply as "G", is to add 1 to it and call "G+1". This is often done as to either say that "there is no largest number, so what's the point" or "look now I HAVE THE LARGEST NUMBER!". As I pointed out earlier, the "point" is simply to keep coming up with larger numbers ... there is no point beyond that. As for the second implication, there is the gentlemen's agreement of googology which basically states "you don't get bragging rights for a naive extension", and that goes double for elementary extensions! Since adding one doesn't present anything new to the system it's a trivial response. And despite the fact that factually speaking ... you have indeed just named the largest number, you'll see that in practice this amounts to nothing. Nobody is ever going to get credit for "coming up with" G+1, because anyone could come up with that. Graham get's credit for Graham's Number however because not everyone could or would bother to. The same goes for almost any googolism. So G+1 is not so much a "new champion" as just a way to say "numbers keep going". Furthermore, rule 34 of googology states "for every googolism, there is a salad number, no exceptions". So rest assured you are not the first person to come along and say "Graham's Number and one". If you think that, your a little late to the party, the line starts waaaaaaaaaaaaaaaaayyyyy back there.

Okay, now that we've got that out of the way ... would you like to go further than G+1? Good.

With that being said, expect some joker, to comment on somebodies comment that they've got an even LARGER NUMBER ... G+2. For a googologist, it's hard not to find this laughable. Of coarse even the most naive would never come up with G+2 as a sincere attempt to extend Graham's Number. It's always meant rather tongue-in-cheek. Same goes for G+3, G+4, G+5, G+6, G+7, G+8, G+9, G+10, G+11, G+12, G+13, G+14, G+15, G+16, G+17, G+18, G+19, G+20, G+21, G+22, G+23, G+24, G+25, G+26, G+27, G+28, G+29, G+30, G+31, G+32, G+33, G+34, G+35, G+36, G+37, G+38, G+39, G+40, G+41, G+42, G+43, G+44, G+45, G+46, G+47, G+48, G+49, G+50 ... etc. This get's pretty silly after a while, and you can guess that someone is probably going to trump all of this and call "2G" as their number! Now, as minute an improvement as this is, it should be noted that it's still an incredible leap forward. Just think of all the numbers we are skipping. Practically a Graham's Number of them! Notice nobody ever bothers to come up with G+100, G+1,000,000, G+10,000,000,000 , G+googol , G+centillion, G+millillion , G+googolgong, G+10^1,000,000, G+10^10^10, G+googolplex, G+googolduplex, G+googoltriplex, G+3^^^3, G+3^^^^3, G+3^^^^^3, G+3^^^^^^3, G+G(2), G+G(3), G+G(4), G+G(5), G+G(6), G+G(7), G+G(8), G+G(9), G+G(10), G+G(11), G+G(12), G+G(13), G+G(14), G+G(15), G+G(16), G+G(17), G+G(18), G+G(19), G+G(20), G+G(21), G+G(22), G+G(23), G+G(24), G+G(25), G+G(26), G+G(27), G+G(28), G+G(29), G+G(30), G+G(31), G+G(32), G+G(33), G+G(34), G+G(35), G+G(36), G+G(37), G+G(38), G+G(39), G+G(40), G+G(41), G+G(42), G+G(43), G+G(44), G+G(45), G+G(46), G+G(47), G+G(48), G+G(49), G+G(50), G+G(51), G+G(52), G+G(53), G+G(54), G+G(55), G+G(56), G+G(57), G+G(58), G+G(59), G+G(60), G+G(61), G+G(62), G+G(63), or ...

G+G(63)+G(62)+G(61)+G(60)+G(59)+G(58)+G(57)+G(56)+G(55)+G(54)+G(53)+G(52)+

G(51)+G(50)+G(49)+G(48)+G(47)+G(46)+G(45)+G(44)+G(43)+G(42)+G(41)+G(40)+G(39)+

G(38)+G(37)+G(36)+G(35)+G(34)+G(33)+G(32)+G(31)+G(30)+G(29)+G(28)+G(27)+G(26)+

G(25)+G(24)+G(23)+G(22)+G(21)+G(20)+G(19)+G(18)+G(17)+G(16)+G(15)+G(14)+G(13)+

G(12)+G(11)+G(10)+G(9)+G(8)+G(7)+G(6)+G(5)+G(4)+G(3)+G(2)+3^^^^^^3+3^^^^^3

+3^^^^3+3^^^3+10^10^10^10^100+10^10^10^100+10^10^100+10^10^10+

10^1,000,000+10^100,000+10^3003+10^303+10^100+10^10+10^6+100+50+49+48+47

+46+45+44+43+42+41+40+39+38+37+36+35+34+33+32+31+30+29+28+27+26+25+24

+23+22+21+20+19+18+17+16+15+14+13+12+11+10+9+8+7+6+5+4+3+2+1

... it's always straight to 2*G. Why is this? The reason is because, the best possible move at that point is to add the largest number currently in play, namely Graham's Number. To the average person, this seems like the really "clever" thing to do. To a googologist however, this is the most fundamental concept, the only thing to really do. If your goal is to name the largest number, and there is a largest number that you currently know, then that is the number you will always use, because it is logically impossible to use anything larger (you can't use a number you don't actually know!). This is the fundamental building block of all googology, and it even has a name ... it's called the recursive step, and when you have an arbitrary number of recursive steps its called a primitive recursion. The primitive recursion is the next most fundamental building block of googology. Thus the "recursive step" is something beyond obvious in googology. So don't expect a single recursive step to get you any significant results in googology circles.

Now there are some interesting things to point out before we move on. First off, despite the fact that it seems like the last number I described less than 2*G, is pretty close to it, the fact is that it is nowhere near it. Even if I had 2*G-G(63) that would still be vastly larger than the massive salad number I created. This seems paradoxical. But let's consider the number ...

G+G(63)+G(62)+G(61)+G(60)+G(59)+G(58)+G(57)+G(56)+G(55)+G(54)+G(53)+G(52)+

G(51)+G(50)+G(49)+G(48)+G(47)+G(46)+G(45)+G(44)+G(43)+G(42)+G(41)+G(40)+G(39)+

G(38)+G(37)+G(36)+G(35)+G(34)+G(33)+G(32)+G(31)+G(30)+G(29)+G(28)+G(27)+G(26)+

G(25)+G(24)+G(23)+G(22)+G(21)+G(20)+G(19)+G(18)+G(17)+G(16)+G(15)+G(14)+G(13)+

G(12)+G(11)+G(10)+G(9)+G(8)+G(7)+G(6)+G(5)+G(4)+G(3)+G(2)+3^^^^^^3+3^^^^^3

+3^^^^3+3^^^3+10^10^10^10^100+10^10^10^100+10^10^100+10^10^10+

10^1,000,000+10^100,000+10^3003+10^303+10^100+10^10+10^6+100+50+49+48+47

+46+45+44+43+42+41+40+39+38+37+36+35+34+33+32+31+30+29+28+27+26+25+24

+23+22+21+20+19+18+17+16+15+14+13+12+11+10+9+8+7+6+5+4+3+2+1

Firstly we can add the last 51 terms easily. Using the triangle number formula we have...

50+49+48+47+46+45+...+5+4+3+2+1

= 50*51/2 = 25*51 = 1275

Now add to this total 100 and we get 1375. Notice that the sum of the first 50 numbers is much larger than 100, so that by adding them all together we get 12 times more than 100. Based on this we can get the naive impression that if we have a long string of smaller numbers then they are bound to add up to a single large number. But that of coarse depends on how large we're talking, and in googology we're usually talking about a number so large that we are probably underestimating it's size. Consider just adding the next term ... 1,000,000. Adding this to the sum of all the other terms is 1,001,375. Now consider this: 1,000,000 plus the sum of all the succeeding terms only increased the number by a measly 0.13%. Now we are actually going to way overestimate the sum as 2,000,000. Do you think this has any chance of significantly effecting the next term? Let's see. The next term is 10^10 or 10,000,000,000. Adding 2,000,000 to that we get 10,002,000,000. Despite the overestimate, this only increased the number by 0.02%. The effect was even less drastic than before! And it get's inconceivably worse from here. Let's again overestimate the sum as 2*10^10. The next term is 10^100. But 10^100+2*10^10 is...

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

,000,000,000,000,000,000,000,000,000,020,000,000,000

At this point the effect of the remaining terms is totally negligible and is practically 0%. Yet we will continue to overestimate by doubling the last term. So we will overestimate this sum as 2*10^100. This number will still be vanishingly small compare to 10^303, and 2*10^303 will still be vanishingly small compare to 10^3003, and 2*10^3003 will still be vanishingly small compare to 10^100,000, and so on through all the terms. The comparisons just get more and more laughable as we go on so that at any point, if we chopped off the last "n" terms, and doubled the new final term, we would still get a number vastly larger! Eventually we would start doing this with the members of the Graham sequence, so that eventually we would have...

G(64)+G(63)+G(62)+G(61)+G(60)+G(59)+G(58)*2

<< G(64)+G(63)+G(62)+G(61)+G(60)+G(59)*2

<< G(64)+G(63)+G(62)+G(61)+G(60)*2

<< G(64)+G(63)+G(62)+G(61)*2

<< G(64)+G(63)+G(62)*2

<< G(64)+G(63)*2

It is easy to forget just how big the gulf between G(63) and G(64) is. Remember, that when numbers are inside functions, the outputs may be vastly different. We can't conclude for example that because 63*2 > 64 therefore G(63)*2 > G(64). Instead, G(64) is vastly vaster than G(63), so that 2*G(63) is still vastly smaller. In fact it's far far worse than this suggests. Consider...

G(64) >> G(63)*100

G(64) >> G(63)*1,000,000

G(64) >> G(63)*googol

G(64) >> G(63)*googolplex

G(64) >> G(63)*(3^^^^3)

G(64) >> G(63)*G(2)

In fact it's even true that...

G(64) >> G(63)*G(63)

We can go much further still. G(63)*G(63) is just G(63)^2. But it's also true that...

G(64) >> G(63)^3

G(64) >> G(63)^10

G(64) >> G(63)^1,000,000

G(64) >> G(63)^googol

G(64) >> G(63)^googolplex

G(64) >> G(63)^(3^^^^3)

G(64) >> G(63)^G(2)

...

G(64) >> G(63)^G(63)

Just think about that for a moment. G(63)^G(63) is still nowhere CLOSE to G(64). G(64) just transcends these elementary extensions of G(63). In fact ...

G(64) >> G(63)^G(63)^G(63)

G(64) >> G(63)^G(63)^G(63)^ ... ^G(63)^G(63)^G(63)

where the number of G(63)'s can be anything you like all the way up to G(63). Understanding this will be the key to appreciating why even naive extensions mop the floor with elementary extensions, and in the long run why a new notation system completely mops the floor with the naive extension of an old notation system.

In any case, try to appreciate the fact that even between G and G*2 is an unimaginable number of numbers. And amongst them will be numbers too complex for us to ever express.

Even this majesty is probably lost on our would-be large-numberers who think G*2 is a pretty neat way to say "see large numbers are easy". So let me clarify: multiplying by 2 is easy, but nobody ever got up to large numbers just by multiplying by 2. It's a naive conclusion to draw that just because a step is easy to carry out, therefore a task is equally easy. Would you consider the fact that you can take a step as any proof that crossing the world on foot, is easy? People assume that because numbers are abstract that they somehow require no labor. This is not true. Did you know that our brain consumes as much as 20% of our energy? Even thought, weighs something. Keep that in mind as we proceed.

Getting back to our ascent to infinity. The next thing most people will do is not try G*2+1. But why? Wasn't adding 1 good just a minute ago? Why is adding 1 suddenly no good anymore. This my friends is how googology begins! You see if this were a competition between several people trying to name a larger number, then everybody already knows that "adding 1" has already been tried. So to pre-empt that, you, intrepid large-numberer, would want to come up with something the others wouldn't already think up based on the game so far, something vastly larger like ... G*3. Of coarse this is also just an elementary extension of Graham's Number, and the problem with elementary extensions is that almost everybody knows what the elementary functions are. So you can rest assured that somebody will probably blurt out something much larger ... G*4 perhaps? Not likely. Since multiplication has been introduced, and G is still the largest number with a compact expression, the next thing you will probably see is G*G.

I'd like to stop here, just to appreciate that this time, we really have made a tremendous leap in terms of the numbers. Human beings tend to think in terms of ratios. We consider something "much bigger" than something else if their ratio is huge. In the case of G*2, G*3, and G*4 the ratio to Graham's Number is a mere 2,3, and 4. In other words, if we could imagine a sphere with the volume of Graham's Number, then G*2 would be a sphere with twice that volume, G*3 would have triple the volume, and G*4 would have quadruple the volume. If these spheres were lines up side by side they would appear to us to be in roughly the same ballpark. But G*G, or simply G^2 is different. Imagine the Graham's sphere next to the G^2 Sphere. The G^2 sphere will have a volume Graham's Number times greater than the Graham's sphere. In other words it would be dwarfed by a factor of itself. It would be worst than a mote of dust on the surface of betelgeuse! In fact we can't even imagine how much bigger it is!! Is it any surprise that people naturally assume that G^2 is so much bigger! But in the wacky world of googology, guess what ... this still doesn't mean a whole hell of beans. From a googologist's point of view a Graham's Number, or G^2 are practically the same thing!!!

To understand this you need to transcend your usual number-sense and start creating your own googological-number-sense. Think of it this way. Imagine beginning from a sphere of volume 1. This sphere is the same relative to the G-sphere as the G-sphere is to the G^2-sphere. Now imagine zooming out from the 1-sphere to the G-sphere, and then zooming out from the G-sphere to the G^2-sphere. Viewed as a journey we would expect to hit the G-sphere at about the half-way point. Thus we can think of G^2 as only having twice the order-of-magnitudes as G. Viewed at this way it only seems twice as large! But how can that be? Because comparisons are relative. They depend on how we are looking at it. Googologist's have ways of looking at it, in which even this comparison sounds generous and G^2 seems no larger than G. How do we see it this way? Well consider beginning with the number 2, the smallest of the large integers. Now square it and you get 4. Now square that and you get 16, now square that and you get 256, now square that and you get 65,536. Keep going. Every time you square the number of digits doubles! Yet this is WAY TOO SLOW to get up to Graham's Number in a reasonable amount of time.

Let's consider some benchmarks. After a mere 9 squares not only do we pass up the number of particles in the observable universe (10⁸⁰) but also a googol (10¹⁰⁰). 10 squares gets us past a centillion (10³⁰³). As you can see we are moving very fast. But how long will it take to reach a googolplex? After 334 squarings we pass up a googolplex. Surely it won't be long before we reach Graham's Number, right? Well guess what it will take about 10³⁵ squarings just to pass up Skewes' Number! And beyond this it will start to seem like the number of squarings is not that much different than the number itself. For example it will take roughly 10^^255 doublings to reach a mega which is approximately 10^^257. Even this small difference will disappear when we get to something like a gaggol (10^^^100). By the time we reach a Graham's Number it will be practically indistinguishable from Graham's Number. Then guess what, you come along and square it one more time, just once, and suddenly it's soooooooooooooooooooooooooooooooooooooo much bigger! Looked at this way, you begin to see things very differently. It seems like squaring Graham's Number is now no better than adding 1! Welcome to the googological number sense! And it gets much much much worse than this, because we can keep inventing new ways to look at the numbers as we progress so that even greater gaps seem negligible. The really scary thing though is, even under such schemes, there is a number so large that even viewed this way it's seems much much larger. We'll get back to the idea of squarings but first let's get back to our progress towards the infinite (surely we must be making progress towards the infinite at this point, right?)

So let's say you've called G^2. If you've written it in this form then you have just introduced exponents into the game. Now anything with Graham's Number and exponents if fair game. Just think of all the numbers we can now construct ... G^3, G^4, G^5, G^6, G^7, G^8, G^9, G^10, G^11, G^12, G^13, G^14, G^15, G^16, G^17, G^18, G^19, G^20, G^21, G^22, G^23, G^24, G^25, G^26, G^27, G^28, G^29, G^30, G^31, G^32, G^33, G^34, G^35, G^36, G^37, G^38, G^39, G^40, G^41, G^42, G^43, G^44, G^45, G^46, G^47, G^48, G^49, G^50, G^51, G^52, G^53, G^54, G^55, G^56, G^57, G^58, G^59, G^60, G^61, G^62, G^63, G^64, ... etc. Now we're getting up to some big numbers. Imagine the G-sphere, and now imagine a sequence of spheres starting with the G^2-sphere,

G^3-sphere, G^4-sphere, etc. Each time we go from one sphere to the next we dwarf the previous sphere by a factor of Graham's Number. Now go up 64 levels ... that is G^64. It sounds almost as impressive as the climb to Graham's Number itself, but remember there is a very big difference between the way the numbers in this sequence dwarf each other and the way the members of the sequence G(1), G(2), G(3) etc. dwarf each other. There is no comparison! In fact if we adopt a googological perspective we see that G, G^2, G^64 are all practically the same number. To reach G^64 we only need to square G an additional 6 times, and remember we had to square G roughly Graham's Number times to get to Graham's Number (more accurately it's actually about loglogG times ). So let's forge on ahead...

G^65,...,G^100,...,G^1000,G^10000, ... G^googol , G^googolplex , ... G^mega ,... G^3^^^3, ... G^G(1) , G^moser , G^G(2) , G^G(3), G^G(4) , G^G(5) , G^G(6) , G^G(7) , G^G(8) , G^G(9) , G^G(10) , G^G(11) , G^G(12) , G^G(13) , G^G(14) , G^G(15) , G^G(16) , G^G(17) , G^G(18) , G^G(19) , G^G(20) , G^G(21) , G^G(22) , G^G(23) , G^G(24) , G^G(25) , G^G(26) , G^G(27) , G^G(28) , G^G(29) , G^G(30) , G^G(31) , G^G(32) , G^G(33) , G^G(34) , G^G(35), G^G(36), G^G(37), G^G(38), G^G(39), G^G(40), G^G(41), G^G(42), G^G(43), G^G(44), G^G(45), G^G(46), G^G(47), G^G(48), G^G(49), G^G(50), G^G(51), G^G(52), G^G(53), G^G(54), G^G(55), G^G(56), G^G(57), G^G(58), G^G(59), G^G(60), G^G(61), G^G(62), G^G(63), ... and lastly G^G.

G^G is likely to be the next number mentioned after G^2, but as you can see there are many many numbers in-between. In fact it's far worse than the above suggests. Remember any member in the above can also be multiplied by any number less than Graham's Number and still be less than the next member, and we can also go ahead and add a number less than Graham's Number. There is a wealth of salad numbers we can form this way. For example...

G^(G(2)*G(1)+3^^^3)*G(17)+googolplex

Even if we included all of the salad numbers we could create this way we would still only account for a minuscule fraction of the total number of integers between a Graham's Number and G^G.

To imagine G^G we have to imagine going along the sphere hierarchy for a Graham's Number of spheres. But what about squaring? Turns out that at this point we finally make a significant dent. Imagine beginning with 2 and continually squaring, once per second. We pass up a googol in less than 10 seconds, a googolplex after 5 and a half minutes, but we won't reach Skewes' number until about ten thousand trillion trillion years! The universe will go completely cold long long before we even reach a mega ... and getting up to Graham's Number is indescribably longer. Yet the journey to Graham's Number will still just be the very beginning of the ascent. Most of the time will be spent trying to reach G^G. So in this sense G^G dwarfs G. Does that mean we finally have a googologically larger number? Noooooo ... of coarse not, we just change our perspective, and now consider a sequence where we take N and go to N^N. In this case we can begin with 2, then have 2^2 = 4, 4^4 = 256, 256^256, then (256^256)^(256^256) , etc. Again we will say this operation occurs every second. Now we pass up a googol after only 3 seconds, googolplex after only 4 seconds, and Skewes' Number after only 5 seconds. What about a mega? That will now take less than 5 minutes to reach! So we'll get to Graham's Number in no time now right? Wrong. Just getting up to 3^^^3 will still take 240,000 years! The universe will become completely cold long long long before we even reach 3^^^4, even at this rate! Getting up to Graham's Number will still take unimaginably long. Once we get up to Graham's Number though, it's just one more second to reach G^G, so in a googological sense, G^G is still virtually the same thing as Graham's Number.

Hmm. So what can we do now? What about factorials. Factorials are stronger than exponents so perhaps that will make a significant dent? Nope. It can be demonstrated that G! < G^G. Why? Because G! is just...

G! = G*(G-1)*(G-2)*(G-3)*(G-4)* ... *4*3*2*1

< G*G*G*G*G* ... *G*G*G*G w/G G's = G^G

So at a certain point the difference between exponential and factorial becomes negligible. In fact we can approximate G! ~ 10^G. This means that G!! ~ 10^10^G , G!!! ~ 10^10^10^G, etc. So even if we write...

G!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

We are still talking about something we could express with exponents. With exponents we can't do much better. Imagine the sequence...

G^G^G , G^G^G^G, G^G^G^G^G, etc.

In fact we can carry this all the way to a power tower of Graham's Number, and Graham's Number terms high. This is commonly cited as a number vastly larger than Graham's Number. It's true that even if we did N-->N^N every second, it would still take longer to go from G to G^^G than to go from 2 to G. But then we can just change our googological perspective and go from N-->N^^N. Again we can begin with 2, go to 2^2 = 4, then 4^4^4^4 , (4^^4)^^(4^^4) , etc.

At this rate things are just insane. We pass up a googol, and a googolplex in a mere 2 seconds. In the 3rd second we pass up little piddling numbers like Skewes' Number, mega, or 3^^^3. 3^^^3, piddling?! Yes we easily pass it up in 3 seconds. How about 3^^^4? In less than 4 seconds. So then it takes 240,000 years to reach something like 3^^^^3? Nope. Much much worse. In 240,000 years even at this rate we only reach about 3^^^3^^3. The universe will go cold long before we reach 3^^^3^^4 and 3^^^^3 would be 3^^^3^^(3^^3). Despite how crazy these analogies are becoming we still can't even come close to bringing Graham's Number down to earth. As you can see, even as we instantly breeze through some very large numbers, there is always just a slightly more sophisticated large number that remains unphased, and Graham's Number lies after a very very long sequence of such numbers. As you can now gather G^^G still occurs 1 second after G at this rate, and it still takes a virtual eternity to reach G.

At this point we have surpassed elementary extensions, because you wouldn't be able to write out G^^G explicitly using elementary functions as in G^G^G^G^...^G^G. Such a power tower wouldn't fit in the known universe! So G^^G is no longer an elementary extension of Graham's Number. This is not surprising because it uses "^^", tetration, a non-elementary function, in order to extend Graham's Number. But it's still a naive extension. Why? Because it uses arrow-notation ... something that was already introduced in the process of introducing Graham's Number. So why would you expect tetration, an operation not even used on G(1) ... to have any affect on Graham's Number? Remember, intuition is no good here. It might seem like 3^^^3 and 3^^^^3 are neighbors. They are not. We will have to go much further than G^^G if we are serious about googology...

Extending Graham's Number with Up-arrow Salad

The next thing people usually do this go head long into Up-arrow notation to extend Graham's Number. As I'll show, no matter what you write out it still has no hope of even getting close to G(65).

Consider the numbers we can create when we include "G" along with arrow notation. If tetration, pentation, hexation, etc. are so powerful, imagine what they could do if G was the argument! It is true that these will lead to massive numbers, necessarily bigger than Graham's Number ... but hold on there ... you ain't seen nothing yet! Consider these examples of numbers beyond Graham's Number ...

G^^G, G^G^^G = G^^(G+1) , G^G^G^^G = G^^(G+2) , ... , G^^(G+100), ... G^^(G+1000) ... G^^(G+googol) ... G^^(G+googolplex) ... G^^(G+googolduplex) ... G^^(G+mega) ... G^^(G+megiston) ... G^^(G+3^^^^3) ... G^^(G+G(2)) , G^^(G+G(3)) , G^^(G+G(4)) ... G^^(G+G) = G^^(G*2) ... G^^(G*3), G^^(G*4) ... G^^(G*10) ... G^^(G*100)...

G^^(G*100) is a power tower of G's 100 times taller than G^^G ... getting pretty big huh. Just wait...

G^^(G*1000) ... G^^(G*googol) ... G^^(G*googolplex) ... G^^(G*googolduplex) ... G^^(G*mega) ... G^^(G*megiston) ... G^^(G*3^^^^3) ... G^^(G*G(2)) ... G^^(G*G(3)) ... G^^(G*G(4)) ... G^^(G*G)...

G^^(G*G) is a power tower of G's a Graham's Number times taller than G^^G. This still is nowhere near where we're headed ...

G^^(G*G) = G^^G^2 , G^^G^3 , G^^G^4 , ... G^^G^100, ... G^^G^googol ... G^^G^googolplex ... G^^G^googolduplex ... G^^G^mega ... G^^G^megiston ... G^^G^3^^^^3 .... G^^G^G(2) ... G^^G^G(3) ... G^^G^G(4) ... G^^G^G ...

G^^G^G is the Graham Numberth power tower in the series G^^G, G^^G^2 , G^^G^3, in which each power tower is a Graham's Number times taller than the previous power tower. This still way too small ...

G^^G^G = G^^G^^2 , G^^G^^3 , ... G^^G^^100 , ... G^^G^^googol , ... G^^G^^googolplex, ... G^^G^^googolduplex ... G^^G^^mega ... G^^G^^megiston ... G^^G^^3^^^^3 ... G^^G^^G(2) ... G^^G^^G(3) ... G^^G^^G(4) ... G^^G^^G ...

G^^G^^G is a power tower of G's G^^G terms high! As awesome as that is ... this is still just the beginning of applying pentation to G. The numbers are getting gigantic aren't they? We'll see ...

G^^G^^G = G^^^3 , G^^^4 , G^^^5 , G^^^6 , G^^^7 , G^^^8 , G^^^9 , G^^^10 , ... G^^^100 ...

To get to G^^^100 begin with G, then have a power tower with G G's, then have a power tower with THAT many G's, and keep going 'til you get up to the 100th member of the sequence. At this point (and well before it in fact) G^^^100 is incomprehensibly larger than G, and this time I'm being serious. The G-sphere is now incomparably small. Does this mean we've finally beaten googology at it's own game ... not even close. Just try N-->N^^^N. Begin with 3. How long do you think it will take to get to G? How long will it take to go from G to G^^^100 ... less than a second ... we've still got a looooooooooong way to go...

G^^^100 ... G^^^1000 , G^^^googol ... G^^^googolplex ... G^^^googolduplex ... G^^^mega ... G^^^megiston ... G^^^3^^^^3 ... G^^^3^^^^^3 ... G^^^3^^^^^^3 ... G^^^3^^^^^^^3 ... G^^^3^^^^^^^^3 ... G^^^G(2) ... G^^^G(3) ... G^^^G(4) ... G^^^G ...

G^^^G is the Graham Numberth power tower! But by this point you probably already know how to go much much further, right? So let's kick it into high gear ...

G^^^G ... G^^^G^G ... G^^^G^^G ... G^^^G^^^G = G^^^^3 ... G^^^^4 ... G^^^^googolplex ... G^^^^G ... G^^^^G^^^^G = G^^^^^3 ... G^^^^^4 ... G^^^^^googolplex ... G^^^^^G ... G^^^^^^G ... G^^^^^^^G ... G^^^^^^^^G ... G^^^^^^^^^G ... G^^^^^^^^^^G ... ... ...

... ... ...

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w/4331 ^s

... ... ...

The last two "massive" numbers are two salad numbers I found on a youtube video about Graham's Number. Surely by the time we reach numbers like this we are getting close to G(65), right? Not even close, as I'll demonstrate shortly.

These are very typical of attempts to extend Graham's Number. Let's consider the first case. We have...

googolplex^^^...387 arrows...^^^googolplex^^^...387 arrows...^^^

G^^^...388 arrows...^^^googolplex^^^...387 arrows....^^^G

The first problem with any expression involving up-arrows and Graham's Number that you could actually write out is ... it only takes 1 more up-arrow to clobber it! Don't believe me? We'll consider replacing all the googolplexes with Graham's Numbers ... that's necessarily much bigger. Now instead of having 387 arrows, replace every case of up-arrows with 388. In that case we have the number...

G^^^...389 arrows...^^^5

This expression will be about 1/5 as long to express and yet it will necessarily be much larger! Hold that thought for a minute. Let's consider the second number...

G^^^...4331 arrows...^^^G

So how close is this number to G(65)? The answer might surprise you, but in order to demonstrate it we will need to be familiar with some of the basic properties of the up-arrow notation.

Say we have an expression of the form a^^^...c...^^^b, where "a" is the base, "b" is the polyponent , and "c" is the Knuth-degree. If we increase either the base, polyponent, or knuth-degree while leaving the others fixed, the result will always be a larger number! I'll borrow Jonathan Bowers' shorthand where, {c} stand in for ^^^...c...^^^. Using this notation we can write the following lemmas for these properties...

LEMMA1 [L1]

a<d --> a{c}b < d{c}b

LEMMA2 [L2]

b<d --> a{c}b < a{c}d

LEMMA3 [L3]

c<d --> a{c}b < a{d}b : a>1 & b>2

Another important property is that a{c}b is always greater than "b"...

LEMMA4 [L4]

b < a{c}b : a>1

The last property is the least intuitive, but the most important for demonstrating the inequalities that will follow...

THEOREM1 [T1]

(a{c}m){c}n << a{c}(m+n) : a>1

In other words, adding the polyponents always results in a larger value (as long as the base is greater than 1). For a thorough demonstration of why these properties are true, see my googology paper, "A Theorem about Knuth-Arrows" (This paper can be found in Appendix C).

Now I will demonstrate that both of these salad numbers get nowhere close to G(65) and in fact there are much much smaller values which are still massively larger than either of them.

First we know that the first salad number is less than

G^^^...389 arrows...^^^5. By Lemma2 this is less than G^^^...389 arrows...^^^G. By Lemma3 this will be less than G^^^...4331 arrows...^^^G, the second salad number, because it has more up-arrows but the same arguments.

It may seem that having Graham's Number as the base and polyponent should make a BIG difference, but it turns out as long as we keep just the polyponent equal to Graham's Number, we will still get a larger number. Firstly we consider...

G^^^...4331 arrows...^^^G

Note that G << 3^^^...4331 arrows...^^^G, by Lemma 4. By Lemma1 we have...

G^^^...4331 arrows...^^^G

< (3^^^...4331 arrows...^^^G)^^^...4331 arrows...^^^G

By Theorem 1 we have that this is less than...

3^^^...4331 arrows...^^^(G+G)

Now consider...

G+G = 2*G < 3*G < 3^G < 3^^G < ... < 3^^^...4331 arrows...^^^G

Thus the previous number is still vastly smaller than...

3^^^...4331 arrows...^^^3^^^...4331 arrows...^^^G

Does this still seem much bigger than Graham's Number? To understand why this number is in some sense, still virtually indistinguishable from Graham's Number we need to consider the construction of Graham's Number. Remember, in arrow notation Graham's Number is just...

3^^^... G₆₃ arrows ...^^^3

or 3{G₆₃}3

When we expand this we get...

3{G₆₃-1}3{G₆₃-1}3

After an inconceivably long time we would eventually expand this to...

3{G₆₃-2}3{G₆₃-2}3{G₆₃-2} ... ... ... {G₆₃-2}3{G₆₃-2}3

with 3{G₆₃-1}3 3's

Because of how up-arrows are defined Graham's Number can be described as a string of 3's and any number of up-arrows between the arguments as long as there is no more than G₆₃ ^s between the 3s. So we can say...

Graham's Number = 3^3^3^3^3^ ... ^3^3^3^3^3 w/N 3's for some very large N

and...

Graham's Number = 3^^3^^3^^3^^ ... ... ^^3^^3^^3^^3 w/N 3's for some very large N

and...

Graham's Number = 3^^^3^^^ ... ... ... ^^^3^^^3 w/N 3's for some very large N

and so on through all the lower hyper-operators.

It is also true that...

Graham's Number = 3^N

Graham's Number = 3^^N

Graham's Number = 3^^^N

Graham's Number = 3^^^^N

and so on through the hyper-operators.

Let's now return to the number...

3{4331}3{4331}G

Remember that G = 3{4332}N for some very large N, and also that G = 3{4331}3{4331}3{4331} ... {4331}3{4331}3. So we have...

3{4331}3{4331}G

= 3{4331}3{4331}3{4331}3{4331}3{4331} ... {4331}3{4331}3

= 3{4332}(2+N)

N is itself a number of the form 3{4332}3{4332} ... {4332}3, so adding 2 to it is insignificant. So we can see that this number is very very close to Graham's Number in this sense. Now consider the following continuation. Imagine that instead of adding 2, we could add any number. We can go all the way up to adding N...

3{4332}(N+N)

But then we can continue to...

3{4332}(N+N+N) = 3{4332}(3*N) < 3{4332}(3^N) < 3{4332}(3^^N)

< 3{4332}(3^^^N) < 3{4332}(3^^^^N) < 3{4332}(3{5}N) < ... ...

3{4332}(3{4331}N) = 3{4332}(3{4331}3{4332}3{4332}...{4332}3)

= 3{4332}3{4332}(1+3{4332} ... {4332}3)

In this way the +1 can climb up this tower of {4332}s. Eventually we reach the top...

3{4332}3{4332}3{4332} ... {4332}3{4332}(3+1)

3{4332}3{4332}3{4332} ... {4332}3{4332}(3+3)

3{4332}3{4332}3{4332} ... {4332}3{4332}(3*3)

3{4332}3{4332}3{4332} ... {4332}3{4332}(3^3)

3{4332}3{4332}3{4332} ... {4332}3{4332}(3^^3)

3{4332}3{4332}3{4332} ... {4332}3{4332}(3^^^3)

3{4332}3{4332}3{4332} ... {4332}3{4332}(3^^^^3)

Eventually we have (3{4332}3) at the top. When this happens we simply reach...

3{4333}(1+3{4333}3{4333} ... {4333}3)

o_0;;;

Eventually this +1 ascends through the tower of {4333}s to form a +1 at the bottom of a tower of {4334}s to eventually ascend the tower of {4334}s to form a +1 at the bottom of a tower of {4335}s and so on!

As you can see it will be a very very very long time before we would make a significant improvement in the size of Graham's Number! Notice that this argument is largely irrespective of how many up-arrows we began with. If you can write it then it's trivially larger than Graham's Number. If we had begun with a 1,000,000 up-arrows it would still be true that G{1,000,000}G < 3{1,000,000}(G*G) <

3{1,000,001}(2+3{1,000,001} ... {1,000,001}3). Regardless of your salad number combination of "G", googolplexes, other miscellaneous googolism's less than G, and up-arrows it will still simply fall somewhere along this hierarchy. In fact even if we went up to 10⁸⁰ up-arrows, enough to fill the universe, we still would be making virtually no progress beyond Graham's Number!!!

But this should really come as no surprise. Remember Graham's Number uses a string of G₆₃ up-arrows. So why would anything less matter?

The Steep Climb to G(65)

Let's consider the climb from up-arrow salad to G(65). As you'll see it's a long climb up. Remember that we can keep ascending a +1 through various levels of power towers. From {4335}-towers we get to {4336}-towers, to {4337}-towers, ... to {1,000,000}-towers to {1,000,001}-towers ... to {10⁸⁰}-towers to {googol}-towers ... to {googolplex}-towers ... to {mega}-towers ... to {megiston}-towers ... to {3^^^^3}-towers.

That's right, eventually we reach G(1) up-arrows, or G^^^....G(1)...^^^G. This number contains far more up-arrows than we could ever hope to write out, but this number is STILL a minuscule improvement on G. This will remain true for a long time. We can imagine the +1 continuing to climb through all the tower structures until we eventually reach G(2) arrows, then G(3) arrows, then G(4) arrows ... after an inconceivably long time we finally begin to be only a handful of up-arrows from having G₆₃ of them. We can imagine just 2 less. Recall that even then the tower is inconceivably tall. Now imagine a single +1 at the bottom...

3{G₆₃-2}(1+3{G₆₃-2}3{G₆₃-2} ... ... ... {G₆₃-2}3{G₆₃-2}3)

with 3{G₆₃-1}3 3's

Eventually the +1 ascends to the next level and we have...

3{G₆₃-1}(1+3{G₆₃-1}3)

Finally the +1's amass to create the massive number...

3{G₆₃-1}3{G₆₃-1}3{G₆₃-1}3

Which is...

3^^^ ... G₆₃ arrows ... ^^^4

where Graham's Number is...

3^^^ ... G₆₃ arrows ... ^^^3

o_0; . . .

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAHHHHHHHHHH!!!

So ALL that work amounts to just growing at the rate of G₆₃ arrows. This means that every salad number is smaller than 3^^^...G(63)...^^^4. So you see, we STILL haven't made much progress. The way forward is NOT to simply take G...G and write a bunch of up-arrows in the middle. The way forward is to come up with higher operators than {G₆₃}.

You may wonder, why can't we just have G^^^...G(63)...^^^G. Isn't that much larger than 3^^^...G(63)...^^^4? Yes. But it's still smaller than 3^^^...G(63)+1...^^^4. What?! Yes. What we are going to find as we progress is that Graham's Number is going to begin to show it's age as we progress to larger and larger numbers. Consider...

G{G63}G = (3{G63}3){G63}(3{G63}3) < 3{G63}(3+3{G63}3) < 3{G63}3{G63}3{G63}3

= 3{G63+1}4

From this point onwards we no longer need to refer directly to Graham's Number to continue. We have...

G = 3^^^...G₆₃...^^^3 < 3^^^...G₆₃...^^^4 < 3^^^...G₆₃...^^^5

<< 3^^^...G₆₃...^^^googolplex

<< 3^^^...G₆₃...^^^(Graham's Number) = 3^^^...G₆₃...^^^3^^^...G₆₃...^^^3

= 3^^^...G₆₃+1...^^^3 < 3^^^...G₆₃+1...^^^4 < 3^^^...G₆₃+1...^^^5

<< 3^^^...G₆₃+1...^^^googolplex

<< 3^^^...G₆₃+1...^^^(Graham's Number)

= 3^^^...G₆₃+1...^^^3^^^...G₆₃...^^^3

At this point even Graham's Number is TOO SMALL an argument!

And we've only just begun ...

3^^^...G₆₃+1...^^^3^^^...G₆₃...^^^3

<< 3^^^...G₆₃+1...^^^3^^^...G₆₃+1...^^^3

= 3^^^...G₆₃+2...^^^3

<< 3^^^...G₆₃+2...^^^(Graham's Number)

<< 3^^^...G₆₃+3...^^^3

<< 3^^^...G₆₃+3...^^^(Graham's Number)

<< 3^^^...G₆₃+4...^^^3

<< 3^^^...G₆₃+4...^^^(Graham's Number)

<< 3^^^...G₆₃+5...^^^3

<< 3^^^...G₆₃+6...^^^3

<< 3^^^...G₆₃+7...^^^3

<< 3^^^...G₆₃+8...^^^3

...

We've only just begun our ascent to G(65). We no longer need to treat Graham's Number as if it were an especially large number. At this scale it is now irrelevant. With this many up-arrows (G₆₃+8 or more) it no longer makes difference whether we use Graham's Number or 3! Weird. The key here is then not to mess around with the arguments, we can leave both the base and polyponent equal to 3, but rather to keep having longer and longer strings of consecutive ^s. This amounts to coming up with a sufficiently large number of arrows. Instead of going at an arrow a time we can jump to stuff like...

3^^^... ... ... G₆₃ + G₆₃ ... ... ... ^^^3

This number has twice the number of up-arrows as Graham's Number! Now there is a large number! Yet we still are nowhere near G(65). We need to keep going...

3^^^ ... ... ... G₆₃ + G₆₃ + G₆₃ ... ... ... ^^^3

or even...

3^^^ ... ... ... G₆₃² ... ... ... ^^^3

then...

3^^^ ... ... ... G₆₃³ ... ... ... ^^^3

then...

3^^^ ... ... ... G₆₃^G₆₃ ... ... ... ^^^3

See a familiar pattern? This was the same way we naively tried to extend G₆₄. For the same reason these naive extensions aren't going to get us very far...

3^^^ ... ... ... G₆₃^^G₆₃ ... ... ... ^^^3

3^^^ ... ... ... G₆₃^^^G₆₃ ... ... ... ^^^3

3^^^ ... ... ... G₆₃^^^^G₆₃ ... ... ... ^^^3

3^^^ ... ... ... G₆₃^^^^^G₆₃ ... ... ... ^^^3

3^^^ ... ... ... G₆₃^^^^^^G₆₃ ... ... ... ^^^3

...

Again the number of up-arrows we could write out will have a negligible effect on G₆₃. Meanwhile written out as a string of up-arrows these expressions are so much longer than Graham's Number expressed in up-arrow notation that it looks like a pin-point in comparison. But we will need much much more up-arrows to reach G(65).

We know that eventually we will go from 3^^^...G₆₂...^^^3 to 3^^^...G₆₂...^^^4 via the climbing method...

3^^^...(3^^^...G₆₂...^^^4)...^^^3

then we have...

3^^^...(3^^^...G₆₂+1...^^^3)...^^^3

See what's happening? The +1 is now burrowing into the Graham's Number recursion! So eventually we would reach...

3^^^...(3^^^...(3^^^...G₆₁+1...^^^3)...^^^3)...^^^3

3^^^...(3^^^...(3^^^...(3^^^...G₆₀+1...^^^3)...^^^3)...^^^3)...^^^3

3^^...(3^^...(3^^...(3^^...(3^^...G₅₉+1...^^3)...^^3)...^^3)...^^3)...^^3

...

Eventually the +1 nests all the way to the very bottom of the barrel...

3^^...(3^^ ... ... ... (3^^...(1+3^^^^3)...^^3) ... ... ...^^3)...^^3

Finally the +1 starts to effect 3^^^^3. We can now imagine a new Graham's Number in which we might begin with something larger than 3^^^^3. For this we can use the following notation.

Let {a,b,c#1} = {a,b,c} = a{c}b = a^^^...^^^b w/c ^s

and...

{a,b,c#d} = {a,b,{a,b,c#(d-1)}}

By this definition we have...

Graham's Number = {3,3,4#64}

Now imagine the number...

{3,3,5#64}

In some sense, starting out with this larger value won't make a big difference. We are still nowhere near G(65). Yet at the same time starting with 3^^^^^3 instead of 3^^^^3 has a ricocheting effect through the entire iteration. Remember 3^^^^^3 is vastly larger than 3^^^^3. Therefore 3^^^...(3^^^^^3)...^^^3 has vastly more up-arrows than 3^^^...(3^^^^3)...^^^3 has, and is therefore incomprehensibly larger! This effect only gets worse as we go down the chain so that by the time we hit the 64th term of each sequence {3,3,5#64} is inconceivably far apart from {3,3,4#64}. Somewhere between these vast numbers is {4,4,4#64}, the Conway-Graham Number we discussed in the previous article.

To get to G(65) we need to go even further. Imagine...

{3,3,6#64} , {3,3,7#64} , {3,3,8#64} ... {3,3,100#64} ... {3,3,1000#64} ... {3,3,1000000#64} ... {3,3,7625597484987#64} = {3,3,{3,3,2}#64} ... {3,3,10¹⁰⁰#64} ... ... {3,3,googolplex#64} ... {3,3,googolduplex#64} ... {3,3,mega#64}

... {3,3,(3^^^3)#64} = {3,3,{3,3,3}#64} = {3,3,3#65} ... {3,3,3^^^4#64} ... {3,3,3^^^100#64} ... {3,3,3^^^googol#64} ... {3,3,3^^^googolplex#64} ...

{3,3,3^^^mega#64} ...

{3,3,3^^^3^^^3#64} = {3,3,3^^^^3#64} = {3,3,{3,3,4}#64} = {3,3,4#65} = G(65).

FINALLY we arrive at G(65). That's the power of Graham's Sequence. With G(65) we trump all the elementary extensions and up-arrow salads the others will come up with. But even G(65) is still considered a naive extension on Graham's Number. Why? Because it still uses a concept introduced by way of Graham's Number. Therefore stuff like G(65) is implicit in the definition of Graham's Number. G(65) does not introduce anything new into the game. If we want to go beyond naive extensions of Graham's Number we'll need to go much further still...

Exhausting the Naive Extensions...

For most people learning about Graham's Number was also the first time they were exposed to up-arrow notation. Once you realize how powerful this notation is it's easy to become quite enamored with it. But Graham's Number also introduces an even more powerful concept ... the G(n) function. So why don't people make more use of the function itself? Some do and some don't. For those that don't there is probably the naive and unverified assumption that... with enough arrows you can easily beat something like G(65). As I have demonstrated however, this naive assumption is false.

It should also be noted that G(65) will also tend to beat an extension on Graham's Number based on anything equivalent to or weaker than Knuth-arrows. Consider the famous xkcd number. This number is defined as...

xkcd = A(G,G)

Where A(m,n) is the 2-argument Ackermann-function. This came from the web-comic with the caption ...

" What does xkcd mean? It means calling the Ackermann function with Graham's Number as the arguments just to horrify mathematicians "

-- xkcd #207

The original web-comic can be found here, in the third panel. Despite the caption, this is definitely not a number that would "horrify" mathematicians. In fact it is easy to show that it is necessarily less than G₆₅. If mathematicians could dream up G₆₄ why would G₆₅ be a problem? It's just the next logical step in a sequence that literally begins at G₁ and continues onwards towards the infinite!

To prove this we simply have to recall the familiar result that...

A(m,n) = 2{m-2}(n+3)-3

So ...

A(G,G) = 2{G-2}(G+3)-3

Now to show this is less than G₆₅ (although googologically speaking very close), observe...

2{G-2}(G+3)-3 < 3{G-2}(G+3) = 3{G₆₄-2}(3+3{G₆₃}3) < 3{G₆₄-2}3{G₆₃}4

< 3{G₆₄-2}3{G₆₃}3{G₆₃}3 = 3{G₆₄-2}3{G₆₃+1}3 <<< 3{G₆₄-2}3{G₆₄-2}3

= 3{G₆₄-1}3 << 3{G₆₄}3 = G₆₅

Therefore...

G₆₄ < xkcd < G₆₅

What about a number like...

G^^^ ... ... ... ^^^G w/G ^s

Technically this number is larger than G₆₅ but googologically speaking ... not by much. In fact even 3^^^...^^^3 w/G+1 ^s is larger. That's right, all we ever will need from this point on to beat arguments of Graham's Number is just a pair of 3's with a single extra up-arrow! Remember ... that even as early as G₆₅ , Graham's Number is a tiny insignificant number, relatively speaking. This may take some getting used to. Remember, no matter how big a number is in googology, 3 seconds later it's completely obsolete. Graham's Number may be super-gigantic the first time we encounter it, but by the time we have this many up-arrows is an irrelevance. We might as well use 3's because 3 and Graham's Number are barely distinguishable even at the scale of G₆₅! Just consider...

G^^^...G...^^^G = (3^^...G₆₃...^^3)^^^...G...^^^(3^^...G₆₃...^^3)

<< (3^^...G...^^3)^^^...G...^^^(3^^...G₆₃...^^3)

< 3^^^...G...^^^(3+3^^...G₆₃...^^3) [Knuth-Arrow Theorem]

< 3^^^...G...^^^3^^...G₆₃...^^4 < 3^^^...G...^^^3^^...G₆₃+1...^^3

Note that the second set of arrows is still vastly smaller than the first. So we can say that...

3^^^...G...^^^3^^...G₆₃+1...^^3

<< 3^^^...G...^^^3^^^...G...^^^3

= 3^^^...G+1...^^^3

So while this number is technically larger than G₆₅ it is still vanishingly small compare to just G₆₆.

So really there is nothing that really compares to the G-function here except something of equal or greater strength (and yes... in googology there are even stronger functions than the G-function, as we will begin to see). Remember that everything that applied to Graham's Number, also applies to it's brethren in the G-sequence. So stuff like...

G₆₅^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^G₆₅

Will still not even come close to G₆₆ and will even be vastly smaller than ...

3^^^^^^^^^ ... ... ... G64 ... ... ... ^^^^^^^^^4

Which is only "slightly" larger than G65. The gulf between each member of the G-sequence is HUGE and only get's larger and larger as we go on. The number of up-arrows involved in each new member of the sequence quickly becomes completely indescribable except in terms of the G-sequence itself. We have...

G(1) = 3^^^^3

G(2) = 3^^^...^^^3 w/3^^^^3 ^s

G(3) = 3^^^^ ... ... ... ^^^^3 w/G(2) ^s

G(4) = 3^^^^ ... ... ... ^^^^3 w/G(3) ^s

G(5) = 3^^^^ ... ... ... ^^^^3 w/G(4) ^s

G(6) = 3^^^^ ... ... ... ^^^^3 w/G(5) ^s

... and this continues to ...

Graham's Number = G(64) = 3^^^^ ... ... ... ^^^^3 w/G(63) ^s

G(65) = 3^^^^ ... ... ... ^^^^3 w/G(64) ^s

G(66) = 3^^^^ ... ... ... ^^^^3 w/G(65) ^s

G(67) = 3^^^^ ... ... ... ^^^^3 w/G(66) ^s

G(68) = 3^^^^ ... ... ... ^^^^3 w/G(67) ^s

G(69) = 3^^^^ ... ... ... ^^^^3 w/G(68) ^s

In mathematics, Graham's Number serves as an upper-bound to a particular problem in Ramsey theory, but in googology Graham's Number holds no special position. It's just a way station, the 64th stop on a train ride to infinity!!!

So let's go much much further. We can go to any stop on this line ... that we can imagine ... and with the G-function in hand ... we can " imagine quite a bit". Consider the awesomeness of a mere G₁₀₀ this number just leaves Graham's Number in the dust! There is just no comparison with G₆₄ anymore ... it's too far down the sequence for that. But from a googological perspective we could say that G₁₀₀ is STILL just down the block from G₆₄. If we are talking about the G-sequence itself, then G₁₀₀ is still, basically, the same idea, just with a slightly larger argument. G₁₀₀ is STILL a naive extension. Why? Because it still doesn't introduce any new concepts. We are just extrapolating from what we learned from Graham's Number. If we want to transcend naive extensions of Graham's Number we will have to go much further.

okay ... how about G(1,000,000). THAT is pretty mind blowing! A million really is a pretty big number. Even if we were counting G₁ , G₂ , G₃ ... it would still take us months and months to count up to G_1,000,000. When you remember that the jump between the members of this sequence are such that each new jump is incomparable to the previous jump you can get kind of dizzy on the largeness ... it's much much much larger than our imaginations can even grasp ... and yet ... we still are just naively extrapolating.

How about G(1,000,000,000), or G(1,000,000,000,000) or G(1,000,000,000,000,000)!

Still not good enough...

Well then G(googol). Now that is VAST. We are beginning to use googologically large numbers as the argument in the G-function. That just leaves the up-arrow stuff in the dust! But just think. How easy is it for someone to clobber your number with something like...

G(googolplex) , G(googolduplex), G(googoltriplex) ... G(mega) , G(megiston) ...

... but wait ... we have something better than any of this ... arrow-notation! So why not try something like G(3^^^3)? This is bigger than G(mega) but smaller than G(megiston).

How about G(3^^^^3)? Now here is a pretty cool number. Eliezer Yudkowsky discusses it when trying to explain how mind-boggling the singularity is ... how we can't understand or even imagine it at our present level of knowledge. Let's call G(3^^^^3) Yudkowsky's Number. It is true that such a number transcends what we have any hope of imagining ... and yet ... we can reach such a number ... even at the present. We can describe it with integer precision with just a handful of simple rules and the notation G(3^^^^3). In this sense we do comprehend it in so far as we understand the processes by which it could in principle be created, if only we had an infinite amount of data and an infinite amount of time. Yudkowsky's point in introducing this number was actually to give some weight to the claim that Graham's Number is probably NOT the largest number at that particular level of complexity! What's complexity? In simplest terms, it's the least amount of information required to describe an idea. Graham's Number may be large by our ordinary standards, but when we adopt a new perspective of "complexity" we see that it is actually not that ambitious. As Yudkowsky showed, one can easily write a computer program to compute Graham's Number. Here's a program to compute Graham's Number in C++:

01 int arrow( int , int , int );

02 int G( int );

03 int main()

04 {

05 return G(64);

06 }

07 int arrow( int base , int prime , int knuth )

08 {

09 size_t interim = 1;

10 if ( knuth==0 )

11 return base*prime;

12 for ( int i = 0 ; i < prime ; i++ )

13 interim = arrow(base , interim , knuth - 1 );

14 return interim;

15 }

16 int G( int number )

17 {

18 if ( number == 1 )

19 return arrow(3,3,4);

20 return arrow(3,3,G( number - 1 ));

21 }

Just 21 little lines of code ... and you have Graham's Number. So what's the catch? You can't even run it right? Yes you can. It will compile without error and run. It crashes and freezes the computer doesn't it? No it runs just fine. It won't look like it's doing anything but behind the scenes its following out the instructions.

The problem isn't the program itself. The problem is memory. The "int" type being used only contains 32-bit numbers which means it can't store numbers greater than 2^32. The program doesn't overflow though. Every time it takes "base*prime" and it exceeds its natural limits it just wraps around. The result? It will continue to chug away, but it's values are almost instantly corrupted by the very tight number-cap. Ultimately the problem is there is just not enough memory in the universe for this program to properly execute, and even if there was there is not enough time for the program to reach completion!

But what we are interested in here is the principle of the thing. The "spirit of the code" if you will. What it's full potential is if the physical restraints were lifted. If you believe that an executable program sitting in your computer memory is just the physical manifestation of an idea and you think that ideas are just realities waiting to happen, then ideas are the next nearest thing to real things. To some extent, in googology, and in mathematics in general, we are asked to take a leap of faith and to believe in the existence of things which only make sense in principle, but which for various reasons may never be practically realized. But the nifty thing is, this faith asks nothing further from us than to invest our imagination in an idea, and the idea returns to us a vision of the sublime and divine. But it does not come without it's cost, for a direct consequence is the inescapable conclusion of unbreachable unknowns ... the eternal torment of any curious mind...

The decimal expansion of Graham's Number is just one such example. As Ronald Graham himself says "I don't know what the first digit of Graham's Number is. Maybe no one will ever know ... but it has one...". To the mathematician, Graham's Number is a real tangible thing, with actual properties just waiting to be discovered. But as soon as we admit Graham's Number into the purview of reality the next thing we quickly realize is that it is almost entirely unknowable. Yudkowsky put it nicely "contained in the digits of Graham's Number is every phone number, every book that has ever been written, or ever could be written, in any encoding you like..." and to go even further I would add that all this would amount to would be just a vanishingly tiny fraction of the total decimal expansion of Graham's Number. Almost everywhere in Graham's Number ... is a digit we will never know! That is not to say we can't know any of it's digits. The last 500 digits of Graham's Number have been computed, but the higher the place value the more computation is required. Eventually something has to give out ... and here's a hint ... it won't be Graham's Number.

So we really are dealing with something quite strange ... a kind of platonic sixth sense about truths of which we have only very very indirect access. And yet as amazing as all that is, there is something really important to understand about that 21 lines of code. It tells us that as BIG as Graham's Number is, it folds up and crumples small enough to fit in your pocket. I'm serious! Think about it. You could write those 21 lines of code in your pocket and take it with you anywhere. Say you were instantly teleported to another dimension in which there was a googologically large computer. In fact the computer is the dimension. It wouldn't even necessarily have to be any more sophisticated than the computers we have today ... it would just have to be sufficiently large, and able to withstand the googological eons of time. It's circuits wouldn't need to be anything more advanced, just the basic logic gates, hell the entire thing could do all it's computation through a little CPU chip no bigger than your thumb. What it would need would be a ginormous memory...

3.2.12