3.2.8
Andre Joyce

 Who is Andre Joyce?!

                    Ironically, despite being perhaps one of the pillars of googology (along with the likes of Kasner, Steinhaus, Conway, and Bowers), very little is known about Andre Joyce by the googologist community. We don't know when he created his googology, we don't know whether he's alive or dead, we don't have any idea what he looked like (the title picture is of Alfred Jerry, not Andre Joyce), we don't even know that he ever existed! There is every indication however that his "work", which goes beyond googology into recreational linguistics and absurdist "mathematics", is obscure even by googologist standards. The only record we have of his existence and googology are extremely dubious texts which describe him as a pataphysicist.

                    This is a term which is an important piece of the Joycian puzzle, for pataphysics is the "science of the absurd", self described as "the science of imaginary solutions to practical problems" ... in other words it espouses to be useless by definition. The purpose and nature of pataphysics is intentionally obscure, but it all boils down to basically flaunting the absurd and irrational. These can be seen in the many texts attributed to Joyce, of which "googology" is only the most cohesive, though far from completely sound.

                    But what is pataphysic's really? Where did it come from? Is it something real or made-up? Self-descriptions aside, how could it be described to an outsider?

                    Pataphysics (sometimes written 'pataphysics) is a 19th century French movement initiated by Alfred Jarry. The term first appeared in print in 1893 and has gained a life of it's own. It has run the gamut from an ideology, a philosophy, a religion, and an important influence on modern art, modernity etc. It should be noted however that we have at least as much to owe to modern science and modern mathematics for our "modern" perspective, and in fact, these developments probably influenced a general sense that the universe is "absurd" and ended up influencing art,philosophy and religion as a result. Although pataphysics predates the 20th century, the 19th century was still a great time of rapid change. There was a general sense of overturning the old and forming the new. Pataphysics was just one of many responses to this, which can be seen all throughout human culture from art, to music, to science, to mathematics.

                    Although I can not go into an in-depth analysis of what pataphysics is, it's influence, and it's purpose(s), some familiarity with it is a prerequisite to separating what I'll call the "pataphysical" instances of Joyce's work from the truly "googological". In a nutshell, the pataphysicist believes that to be human is to be "absurd", in that the human seeks meaning where there is none. To relieve the pressure from the contrast between reality and our humanity, pataphysicists seek to be more like the universe by embracing the nonsensical, making it near impossible to find meaning. Thus by being more "absurd" in human terms they are actually being less absurd in reality (I have plenty of philosophical qualms with this "solution" but I can not go into an in-depth explanation here) Therefore to try and understand a pataphysical work rationally is to miss the point.

                    Googology on the other hand is not about exploring the nonsensical, however much it appears to the contrary on the surface. Googology is about exploring a rational world of numbers made of mathematical building blocks. 

The pataphysical and the googological are fundamentally at odds, and it shows in the text. It is both a mix of reason and nonsense.

                    Googologist's express the "absurd", and the whimsical, in the way they name numbers to express the sense of dumbfoundedness that these numbers produce. They lie beyond human capacity to understand and therefore become emblematic of all that we yearn for and fear ... a god-like realm filled with unfathomable sound and fury and yet completely without purpose, ferocious yet without malice, infinitely ingenious and yet without a mind to appreciate their ingenuity. As was said of Turing Machines "if intelligence lies in their design, no mind put it there". In short, the googologist is every bit in touch with the pataphysicist's attempt to communicate with the divine realm of the absurd, but the googologist recognizes that the most important characteristic of this realm is not the irrational ... but the hyper-rational. The universe follows it's own laws, one might say, maniacally, without concern of what the consequences might be and therefore without "purpose". This is what we human beings react to as "absurd". The response however is not to descend into madness ... but to seek to understand the order while retaining one's sense of purpose. By delving into and toying with madness Joyce invites sloppy thinking, and it shows in Joyce's work.

                    With that in mind, I will point out the places where I believe pataphysics is to blame. There are some places that are so erroneous in the text, that Joyce has to be kidding (then again, maybe not. It's impossible to tell!)

The Text

                    The "text" of which I speak is actually a secondary source. It is claimed to be drawn from "Let Us Remember Andre Joyce" but there has never been any evidence to support that such a text exists. A google search of the name will just return the "secondary sources". To make matters worse, there are two texts which are very different but seem to be talking about the same Andre Joyce. One text, is said to be a translation of Andre Joyce's "Googology" by Michael Joseph Halm [1]. Another text, is said to be translated by Razilee Purdue [2]. The Razilee Purdue article predates the Michael Joseph Halm article. Razilee Purdue's article is dated 1978-2003 while Michael Joseph Halm's is dated 2006. The texts are actually extremely different and have few commonalities. One curious thing to note is the Razilee's article does not mention any Bowerisms (googolism's invented by Jonathan Bowers'), but Michael's does. Michael's appears to be an expansion of Razilee's. The fact that each article contains unique material further supports the idea that there really is no original text. Both text's lack any organization. They mostly just introduce a bunch of random constructions with very little introduction. Not much additional information can be found on Andre Joyce or his googology. Something of a profile of Andre Joyce can be found at [3]. The texts on googology, don't actually introduce Joyce's prevalent g-notation very well, but some additional clues can be found at the bottom of this page at [4].

                Beyond that there is literally no more information to go on to decipher Joyce's googology. All other sources are simply copied and pasted from these texts, including their errors. Joyce's googolism's often appear along side Bowerism's. A typical example can be seen here [5]. One consistent problem is that Joyce's numbers are listed as larger than some of Bowers', yet it turns out that they are no where near them and are much much smaller! This was the result of a misunderstanding of whoever prepared the text (Joyce or Michael, it can't be known for sure), and it has unfortunately been copied verbatim into many other sources. For the first time, I will explore this claim and show it to be false. Let's begin.

The Ackermann Function

                    Before we get to Joyce's googology, let's first familiarize ourselves with a notation which he uses heavily and calls the "generalized Ackermann Function". This function turns out to be just Knuth's up-arrows in disguise. A three-argument function is used appended with a "g". The arguments are in decreasing significance, so that the first argument is the most significant, following by the second and the third. The first argument we can call the "degree", the second argument the "polyponent", and the third argument the "base". It is generally written in the form:

g(a,b,c)

                    The following loose definition is provided ...

g(a, b, c) = g(a - 1, g(a, b - 1, c), c),

g(0, a, b) = a + b,

g(1, a, b) = ab

                    The problem with this is it doesn't actually provide a proper terminating case. if the first argument = 0 or 1, then we have no problem and can simply take the sum or product of the remaining arguments respectively. But if the first argument >1 then we have to apply the first rule. Here we have an example ...

g(2,3,3) = g(1,g(2,2,3),3)

We could add this if only we knew what g(2,2,3) was. To find out we again expand it using rule 1 ...

g(1,g(2,2,3),3) = g(1,g(1,g(2,1,3),3),3)

the problem is we then need to know g(2,1,3). Using the rule we have g(1,g(2,0,3),3) ... but then we need to know g(2,0,3). To know g(2,0,3) we need to know g(2,-1,3), and to know g(2,-1,3) we need to know g(2,-2,3) ... and then in turn g(2,-3,3), g(2,-4,3) , g(2,-5,3), etc.

                    What's missing is a rule which tells us when to stop. It's clear from the examples that follow however that g(2,a,b) = b^a. From this we can gather that g(2,1,b) = b^1 = b

and g(2,0,b) = b^0 = 1. For the sake of consistency we can use this later rule, since the first argument can range over the non-negative integers, then the second argument should also range over the non-negative integers. Thus in our above example we should get...

g(2,3,3) = g(1,g(2,2,3),3) = g(1,g(1,g(2,1,3),3),3) = g(1,g(1,g(1,g(2,0,3),3),3),3)

Since g(2,0,3) = 1 by definition we get...

g(1,g(1,g(1,g(2,0,3),3),3),3) = g(1,g(1,g(1,1,3),3),3)

Next we apply the 3rd rule ...

g(1,g(1,g(1,1,3),3),3) = g(1,g(1,1*3,3),3) = g(1,g(1,3,3),3) = g(1,3*3,3) = g(1,9,3) = 9*3 

= 27

                    This omission is somewhat justified, on account of the fact that this is just the Ackermann function with the arguments in reverse. Ackermann originally defined his 3-argument function as [6] ...

φ(m,n,p) where m,n,p are elements of {0,1,2,...} :

φ(m,n,0) = m+n

φ(m,0,1) = 0

φ(m,0,2) = 1

φ(m,0,p) = m : for p>2

φ(m,n,p) = φ(m,φ(m,n-1,p),p-1) : n,p>0

                    This closely agrees with treating "p" as representing the number of up-arrows plus one. Furthermore we find that ...

φ(m,n,0) = m+n

φ(m,n,1) = m*n

φ(m,n,2) = m^n

                    However, due to the 4th rule we find that...

φ(m,n,3) = m^^(n+1)

φ(m,n,4) = m^^^(n+1)

φ(m,n,5) = m^^^^(n+1)

etc.

                    We will use a more modern definition which agrees with the hyper-operators without this offset. To do this the rules must be changed slightly...

g(c,b,a) where c,b,a are elements of {0,1,2,3,...} :

1. c=0 ; g(0,b,a) = b+a

2. b=0 , c=1 ; g(1,0,a) = 0

3. b=0 ; g(c,0,a) = 1

4. g(c,b,a) = g(c-1,g(c,b-1,a),a)

                    These rules are prioritized, so it is not necessary to state for rule 3 that c>1. It's implied since rule 3 can only apply if b=0, but if a=0 then rule 1 would apply first, and if a=1 then rule 2 would apply. Therefore Rule 3 can only apply if c>1. Furthermore it's not necessary to specify that b and c both be greater than 0 in rule 4. If c is 0 then regardless of the value of b rule 1 would apply. If b is 0 and c is not then rule 2 or 3 applies. Ackermann did not use prioritized rule sets but rather used the piece-wise function notation. Prioritized rulesets was pioneered by Bowers'. The technique has proven to be a useful general approach to defining recursive functions and is a common standard used by the googology community.

                    In any case this should eliminate any ambiguity in regards to how the g(c,b,a) function should work, even if not explicitly stated. In general we can say that...

g(0,b,a) = b+a

g(1,b,a) = b*a 

g(2,b,a) = a^b

g(3,b,a) = a^^b

g(4,b,a) = a^^^b

g(5,b,a) = a^^^^b

and in general

g(c,b,a) = a^^^...^^^b w/c-1 ^s for c>1

alternatively it is convenient to use multiple stars where...

* denotes multiplication

** denotes ^

*** denotes ^^

and in general

***...*** w/n *s denotes ^^^...^^ w/n-1 ^s

Thus...

g(c,b,a) = a***...***b w/c *s : c>0

Joycian Googology

                    With that cleared up we can proceed further with Joyce's googology.

                    It begins with the observation that there are many ways to describe the googol using "Ackermann's Generalized Exponential Notation" (shortened to AGE from here in). 

                    The equality ...

googol = g(2,100,10) = g(2,2,g(3,2,10)) = g(2,50,g(1,2,50))

                    ...is given. The first expression is clearly equal to a googol since g(2,100,10) = 10^100. The other expressions seem convoluted and it's less obviously true. Looking at the 2nd expression and simplifying we have...

g(2,2,g(3,2,10)) = g(2,2,10^^2) = g(2,2,10^10) = (10^10)^2

                    and here we encounter one of the first anomalies. In one of the sources it is claimed that Andre Joyce is a "Franco-American mathematician", but it is clear from the equality...

10^100 = (10^10)^2

                    that we are not looking at the work of a professional mathematician, for this is an egregiously elementary error! Naively it is often assumed that:

(a^b)^c = a^(b^c)

                    This is not true in the general case. Take for example...

(3^3)^3 = (27)^3 = 19,683

while ...

3^(3^3) = 3^27 = 7,625,597,484,987

                    Typically we find that (a^b)^c << a^(b^c). Furthermore it turns out that...

(a^b)^c = a^(b*c)

                    This is, of coarse, one of the basic exponential laws, and yet Andre has somehow got it wrong! Given that pataphysics is the science of the absurd, is it knowingly false, or false out of ignorance? I can't say. But it doesn't matter. Knowingly or unknowingly it's wrong. The actual evaluation would be...

g(2,2,g(3,2,10)) = (10^10)^2 = 10^(10*2) = 10^20 != 10^100

                    We will have to be wary for further cases of poor analysis. Is the 3rd expression correct?

g(2,50,g(1,2,50)) = g(2,50,2*50) = g(2,50,100) = 100^50

Yes. 100^50 = (10^2)^50 = 10^(2*50) = 10^100. It is strange that Andre get's this right, but somehow get's the other one wrong. A simple correction would be...

g(2,10,g(2,10,10)) = (10^10)^10 = 10^(10*10) = 10^100

                    yet this is not what we see. Andre goes on to define the googolplex as ...

googolplex = ten-to-the-googolth = g(2,g(2,100,10),2) = g(2,2,g(3,3,10))

                    Inexplicably only "ten-to-the-googolth", is a correct definition! g(2,g(2,100,10),2) would be 2^10^100 not 10^10^100. 2^10^100 is smaller than 10^10^100 ( 2^10^100 ~ 10^10^99.5 ). Again a simple correction ...

googolplex = g(2,g(2,100,10),10) = 10^10^100

                    Why the base of 2? To make matters worse we can see that...

g(2,2,g(3,3,10)) = g(2,2,10^^3) = g(2,2,10^10^10) = (10^10^10)^2

This demonstrates a profound ignorance of how large numbers actually work! The implication here is that...

(10^10^10)^2 = 10^10^10^2 = 10^10^100

This is completely erroneous! The correction evaluation is...

(10^10^10)^2 = (10^(10^10))^2 = 10^((10^10)*2) = 10^(2*10^10)

Instead of giving us 1 followed by a googol zeroes, it reduces to 1 followed by a mere 20 billion zeroes, which is vastly smaller. Even the decimal expansion would be dwarfed relative to the decimal expansion of a googolplex. The difference in magnitude is incomprehensible and is indistinguishable from a googolplex. In general when A/B ~ A in googological terms it means that A is vastly larger than B. This is typically not the exception in googology, but the norm. Numbers are rarely close enough in magnitude that A/B produces a comfortably sized number (astronomical or smaller), and the ratio is usually itself googologically large. For the person who was the first to call himself a "googologist", Andre sure doesn't know his exponential laws!

                Viewed from the outside, googology (as practiced by the googology community) may seem indistinguishable from pataphysical nonsense. "Imaginary solutions" of impractical value. It may be true that googology is impractical, but it MUST be distinguished from nonsense if it is to have any value at all. The point of googology is to invent well-defined large numbers, and to be able to compare various large numbers through a sound mathematical analysis. Let it not be inferred from the whimsical names that googologist's often give to the numbers they define, that there definitions are also whimsical! If it is not well-defined it's not a number!

                    From this point of view, it might be argued that what Andre Joyce was doing wasn't quite "googology" as we recognize and understand it today. He may have been largely unconcerned with how correct he was. The entire concept of "googology" was probably something of a joke. It's absurd as a study. So he perhaps gave it the same irreverent focus as his "jootsy calculus" and other profoundly ridiculous things. The difference with his googology however was that it was the closest thing to sense.

                    For the sake of posterity we will continue to go through Andre's work, but we have more than fair warning that there are problems already.

                    After the prefunctory introduction, Andre finally get's down to naming his first googolism:

eleventyplex = g(2,110,10) = 10^110

This is perfectly sound, if a bit silly. Next however we have...

millilliardplex = g(2,6003,g(3,2,10))

We know from our exploration of -illion schemes that a millilliard is from the long scale system in which a millillion would be 10^6000, and therefore a millilliard would be 10^6003. Therefore a millilliardplex should be 10^10^6003 based on the established conventions. Yet we have...

g(2,6003,g(3,2,10)) = g(2,6003,10^^2) = g(2,6003,10^10) = (10^10)^6003 

which is 10^60,030 not 10^10^6003 which is MUCH larger. At least Andre is wrong in a consistent way. But we should know better. Remember (a^^b)^c != a^a^...^a^c w/b a's. Instead we have (a^^b)^c = (a^a^^(b-1))^c = a^(c*a^^(b-1)) which is generally much smaller. 

Given this consistent mistake we can assume that whenever Andre uses (a^b)^c what he is actually trying to summon is the number a^(b^c). The key to getting these values is to nest the 2nd argument of the g-function, NOT the first! So it should be...

millilliardplex = g(2,g(2,6003,10),10)

                This still doesn't really establish anything new. It's a well-established practice that n-plex = 10^n. The next idea however is new.

                Joyce observes that gross = 12^2 and great gross = 12^3. So he concludes that if n = g(2,a,b) then great-(n) = g(2,a+1,b). So a gross is g(2,2,12) and a great gross is g(2,3,12). This is a very curious observation and it proves to not even be followed anyway. By such reasoning, since googol = g(2,100,10), a great googol should be = g(2,101,10). Instead great googol = g(2,1000,10) ... so much for any kind of consistency -_-

                Yet there is a vague amount of sense here since googol = 10^10^2 and great googol = 10^10^3. So a rule that does more closely match what is happening would be ... if n = f(n), then great-(n) = f(n+1). In this case let f(n) = 10^10^n, googol = f(2) and great googol = f(3).

                Joyce goes on to define...

great googolplex = g(2,1000,g(3,2,10)) = (10^10)^1000

which should actually be g(2,g(2,1000,10),10) = 10^10^1000

                    He goes on to generalize this to...

n-ex-great googol = g(2,n+2,g(2,100,10)) = (10^100)^(n+2)

n-ex-great googolplex = g(2,n+2,g(2,100,g(3,2,10))) = ((10^10)^100)^(n+2)

                    The degree of brokenness here is quite unbelievable. Even by Joyce's already tortured arithmetic standards, this definition makes no sense. Let n=1. For great googol we get (10^100)^(1+2) = (10^100)^(3). Even if we adopt his broken logic we would have 10^100^3, but this is NOT 10^1000. Instead it would be 10^1,000,000. Furthermore the actual value would be (10^100)^(3) = 10^300. To get the result he is going for he needs 10^10^(n+2), which can be achieved using g(2,g(2,n+2,10),10). Likewise for n-ex-great googolplex what is needed is 10^10^10^(n+2), which can be achieved using g(2,g(2,g(2,n+2,10),10),10). Viewed at this way Joyce's suggestion is actually somewhat useful for filling out the spaces between the members of the googol series. For example we have...

great googol = 10^1000

great great googol = 10^10,000

great great great googol = 10^100,000

great great great great googol = 10^1,000,000

great great great great great googol = 10^10^7

great great great great great great googol = 10^10^8

etc.

great googolplex = 10^10^1000

great great googolplex = 10^10^10,000

great great great googolplex = 10^10^100,000

great great great great googolplex = 10^10^1,000,000

great great great great great googolplex = 10^10^10^7

great great great great great great googolplex = 10^10^10^8

etc.

                    This is actually somewhat useful, as these numbers can act as various hyper-orders of magnitude. For example we can say 3^3^3^3 is roughly of the order of 10^10^12. In Joycian googology this would be a ten-ex-great googol. 

                    Googology typically involves numbers larger than a googol but occasionally the googological impulse can also involve smaller numbers, since part of googology is about inventing names for numbers. Joyce extends the great gross with ...

great great gross = 12^4

great great great gross = 12^5

great great great great gross = 12^6

great great great great great gross = 12^7

great great great great great great gross = 12^8

etc.

                    using the formula 'n-ex-great gross' = g(2,n+2,12). For once he get's it right! This works nicely and is a fairly natural extension.

                    Going further, since a Baker's dozen is 13, and a Baker's gross is 169 or 13^2, it follows that a great Baker's gross should be 2197 or 13^3, that a great great Baker's gross should be 13^4, etc. and that in general 'n-ex-great Baker's gross' = g(2,n+2,13). Unfortunately a typo in the article states 'n-greats Baker's gross = g(2,n+2,12).

                   Going further still, we have Poulter's dozen = 14, Poulter's gross = 196 = 14^2, great Poulter's gross = 14^3, and in general n-ex-great Poulter's gross = g(2,n+2,14). This time there is no typo.

                    After this more googolism's follow, this time without explanation...

great googol = g(2,3/2,g(2,100,10)) = (10^100)^(3/2)

here we get a 4th value for great googol, in addition to 10^101, 10^300, and 10^1000. This time great googol = 10^150. What Joyce means, of coarse is, great googol = 10^100^(3/2) = 10^1000. We see here that this exponential error is systemic and is not an isolated incident.

                    Moving on we have...

googolteen = g(2,100,10)+10 = 10^100+10(from googol+teen )

googolty = g(2,101,10) = 10^100*10 = 10^101 ( from googol+ty)

googolilliard = g(2,6*g(2,2,100,2)+3,10)

                    This use of a 4-argument g-function is probably another typo. It is probably meant to be g(2,6*g(2,100,10)+3,10) = 10^(6*10^100+3) which is consistent with n-illiard being 10^(6n+3).

googolylliard = g(2,16*g(2,100,10)+8,10) = 10^(16*10^100+8)

googolplex = g(2,g(2,100,10),10) = g(2,g(2,g(2,2,10),10),10) = g(3,2,2,10)

eleventyplex = g(2,110,10)

                    The construction of googolylliard appears to be using Knuth's -yllion suffix. In Knuth's system each new denomination is the square of the previous one [7]. He begins with a myriad (10,000) as the base, and then let's myllion be myriad^2 or 10^8. From here on Knuth defines...

(n)-yllion = 10^2^(n+2)

                    By this reckoning googolyllion should be 10^2^(10^100+2), but -ylliard is not defined in Knuth's system, so the meaning of googolylliard is unclear. It appears Joyce has made the mistake that Knuth's system uses 10^16 as the base multiplier so that , myllion = 10^16, byllion = 10^32, tryllion = 10^48, etc. and therefore n-yllion = 10^(16n) while n-ylliard = 10^(16n+8). This is a misunderstanding of how Knuth's system works, but at this point I'm not too surprised.

                    The more interesting case is the new equalities for a googolplex which appear to suggest a way to continue to a 4-argument g-function. The first part is correct. googolplex = g(2,g(2,100,10),10) = g(2,10^100,10) = 10^10^100. As for the second part we have...

g(2,g(2,g(2,2,10),10),10) = g(2,g(2,10^2,10),10) = g(2,g(2,100,10),10) = 10^10^100

So this is also correct. Lastly we see that...

g(3,2,2,10) = g(2,g(2,g(2,2,10),10),10)

This suggests the definition...

g(1,a,b,c) = g(a,b,c)

g(n,a,b,c) = g(a,g(n-1,a,b,c),c)

                    While this definition is well-defined (although only implicit!) it is nothing particularly remarkable, and in fact is poorly structured for making larger numbers. First and foremost, the new argument is simply acting as a functional power, feeding back into the 2nd argument. This technique is known as primitive recursion. Primitive recursion is a very basic technique, and it was defined prior to 1928. So this is nothing new. To make matters worse, the primitive recursion should be applied to the 1st argument not the 2nd! The reason is that, this will make the new argument redundant. You see, by recursively plugging into the 2nd argument, we are doing exactly the same thing that the 1st argument already accomplished. Therefore we will find that g(n,0,a,b) grows just as fast as g(1,n,b) , g(n,1,a,b) grows just as fast as g(2,n,b), that g(n,2,a,b) will grow just as fast as g(3,n,b), and in general g(n,k,a,b) will grow just as fast as g(k+1,n,b) (plus a constant of offset). Don't believe me? Consider the following sequence.

g(3,1,10)

g(3,2,10)

g(3,3,10)

etc.

vs.

g(1,2,10,10)

g(2,2,10,10)

g(3,2,10,10)

etc.

                    In the first case we get g(3,1,10) = 10 , g(3,2,10) = 10^^2 = 10^10 , g(3,3,10) = 10^^3 = 10^10^10 , g(3,4,10) = 10^10^10^10, etc. On the other hand we have...

g(1,2,10,10) = g(2,10,10) = 10^10

g(2,2,10,10) = g(2,g(1,2,10,10),10) = g(2,g(2,10,10),10) = 10^10^10

g(3,2,10,10) = g(2,g(2,2,10,10),10) = g(2,10^10^10,10) = 10^10^10^10

As you see we are describing the same sequence, except that the second sequence has a offset of 1. A function grows faster than another if an only if, it can always eventually surpass it no matter how much of a headstart the other function gets. If however the functions can't catch up to each other if the other is given a lead, then neither is actually faster than the other and they are growing at the same rate. This is the case here. So Joyce has haphazardly created a redundant recursion. If we recursively plugged into the first argument however, this would produce a radical new growth rate. 

As it turns out later in the text, this assumption proves false. We find that the new 4th argument (which is listed 1st), actually iterates for the last argument not the second to last. Rather it is the 5th argument that iterates the 2nd to last. In light of this,

googolplex = g(3,2,2,10)

is probably a typo and what was meant was

googolplex = g(3,1,2,2,10)

... or maybe Joyce was just being patalogical ...

Googo- Operator

                    Oddly, rather than follow the logic and try to create a powerful polyadic g-function, Joyce instead opts to go back to a googol and make a very curious observation and than extrapolate from that.

                    Andre Joyce has become part of the folk-lore and history of the googology community. Yet little is known about his work, and even less of the man himself. As a sort of cornerstone of the idea of continuing where Milton and Kasner left off, he holds a certain respect as a forefather. His work has been copied countless times on other sites of large number enthusiasts, and his claims have mostly been taken at face value. But how large are the numbers Joyce came up with? How precisely does his notation work? Is it even properly defined? How relevant is Joyce's work in today's "googology"?

                    "Googology" has gained a life of its own, gaining in sophistication and moving from mere humor to something pursued with a "seriousness" that probably would have thrown Joyce off!

                    In this article we will take a critical look at the seminal text of Andre Joyce, a sacred cornerstone of googology ( a google search of "googology" will bring it up on the first page right with googology.wiki, and that's saying something). In so doing some of mystique may wear off. There is nothing more impressive sounding than a vague idea! But I promise that the lessons we will learn from it will make us all much better googologist's in the long run. There is little sense to this senseless subject of googology if it is not at least mathematically sensible!

                    Ironically, despite the familiarity the googological community has with Joyce's work, very little analysis has ever been brought to bare on it on the internet. I seek to remedy this forthwith. This is in part because the text itself is confusing, vague, and poorly written. I myself had avoided giving it a proper analysis because it didn't look particularly powerful, but it did look particularly confusing! So I put it off for many years. Having finally faced the text head on, my suspicions have been confirmed. None of the ideas presented in Joyce's work are revolutionary, and the text is vague, confusing and worse yet contains many many errors.

                    Why bother then? Because there are lessons to be learned, distinctions to be made, and viable googolism's to harvest. I do believe the text does capture something of the spirit of the modern googology movement, the idea of creating large numbers without some practical end in mind (this is in contrast to things like Graham's Number, where it is stressed that the number is connected to some problem in serious mathematics). Joyce's googolism's, such as a googoc, for better or worse, have become part of the developing canon of googology, and they can be found along side those googolism's invented by Bowers' and others and various lists of googolism's. I would be remiss to simply ignore Joyce's work. It is simply too prevalent to ignore. 

                    There is also one last very important reason to take a critical look at Joyce's work. Joyce's claims of reaching some of Bowers' higher googolism's has needed to be debunked for a long time. Joyce's claims, as I shall demonstrate with support of the text, are completely erroneous. Unfortunately this has not stopped non-googologist's from taking his claims at face value. I will seek to rectify this manner.

                    In short, I seek to once and for all set the record straight on Joyce's work; the good, the bad, ... and the down right patalogical ... :/

mag-(n) = g(n,n+1,n,2)

and that Joyce continued it further with ...

bag-(n) = g(2,n,n+1,n,2)

trag-(n) = g(3,n,n+1,n,2)

quadrag-(n) = g(4,n,n+1,n,2)

From these various prefixes Joyce suggests the formations fugagoogol , gaggoogol , and megafugagoogol, but he only provides an explicit definition for the last one...

megafugagoogol = g(4,2,g(2,100,10)) = (10^100)^^^2 = (10^100)^^(10^100)

which is, thankfully, correct. He goes on further to suggest that...

baggoogol = g(2, z, z + 1, z, 2), traggoogol = g(3, z, z + 1, z, 2), quadraggoogol = g(4, z, z + 1, z, 2), etc.

--[sic] Joyce

umm ... what is "z"? As you can see the text has this really bad and (noticeable) tendancy to copy and paste text without making the necessary adjustments. We can surmise that "z" was suppose to be substituted with googol. In this case we should have...

baggoogol = g(2,10^100,10^100+1,10^100,2)

traggoogol = g(3,10^100,10^100+1,10^100,2)

quadraggoogol = g(4,10^100,10^100+1,10^100,2)

These are 5 argument cases. Applying the appropriate rule we can actually compute these values. We have...

baggoogol = g(2,10^100,10^100+1,10^100,2) = g(1,10^100,10^100+1,g(10^100+1,10^100,2),2) =

g(10^100,10^100+1,g(10^100+1,10^100,2),2)

The problem: we run smack dab into the 4-argument case. If we take Joyce's rules at face value than a gear jam is inevitable and we will find this is no greater than...

g(10^100+1,g(10^100+1,10^100,2),g(10^100+1,10^100,2)^(10^100+1))

via...

g(a,b,c,d) = g(b,c,c^b)

 A confusing mess, no? To make matters simpler we can approximate it as follows...

g(10^100+1,g(10^100+1,10^100,2),g(10^100+1,10^100,2)^(10^100+1))

~ g(10^100+1,g(10^100+1,10^100,2),g(10^100+1,10^100,2))

= (2^^^...^^^googol)^^^...^^^(2^^^...^^^googol)

w/^^^...^^^ contains googol ^s

That's better. Now we can use the Polyponent-Addition-Lemma (PAL) to say...

(2^^^...^^^googol)^^^...^^^(2^^^...^^^googol)

~ 2^^^...^^^(googol+2^^^...^^^googol)

~ 2^^^...^^^2^^^...^^^googol

... This is a huge number to be sure, but it's mainly due to the fact that we literally have a googol up-arrows between the arguments. So what about a traggoogol? We'll in that case we have...

traggoogol = g(3,10^100,10^100+1,10^100,2) =

g(2,10^100,10^100+1,g(10^100+1,10^100,2),2) =

g(1,10^100,10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),2)

= g(10^100,10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),2)

again the gear jam in rule 1 means having a googol in the 4th argument makes no difference. We get ...

g(10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),g(10^100+1,g(10^100+1,10^100,2),2)^(10^100+1))

~ (2^^^...^^^2^^^...^^^googol)^^^...^^^(2^^^...^^^2^^^...^^^googol)

~ 2^^^...^^^(2^^^...^^^googol+2^^^...^^^2^^^...^^^googol) [PAL]

~ 2^^^...^^^2^^^...^^^2^^^...^^^googol

Although the number of up-arrows is very large, (seriously this wouldn't fit in the observable universe!), we are still only iterating the next operator (the googol+1 up-arrow operator). This is a little unusual for numbers this big. Usually once we have this many arrows we'd want to start iterating the number of up-arrows. We can easily between baggoogol with

g(googol+2,5,2) and traggoogol with g(googol+2,6,2). So we aren't exactly pushing the limits of the Ackermann function here. It's almost as if we are going sideways when we should be going up. None the less, let's finish with the quadraggoogol ...

quadraggoogol = g(4,10^100,10^100+1,10^100,2) =

g(3,10^100,10^100+1,g(10^100+1,10^100,2),2) =

g(2,10^100,10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),2) =

g(10^100,10^100+1,g(10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),2),2)

= g(10^100+1,g(10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),2),g(10^100+1,g(10^100+1,g(10^100+1,10^100,2),2),2)^(10^100+1))

~ (2^^^...^^^2^^^...^^^2^^^...^^^googol)^^^...^^^(2^^^...^^^2^^^...^^^2^^^...^^^googol)

~ 2^^^...^^^(2^^^...^^^2^^^...^^^googol+2^^^...^^^2^^^...^^^2^^^...^^^googol)

~ 2^^^...^^^2^^^...^^^2^^^...^^^2^^^...^^^googol

< 2^^^...^^^^7 = g(googol+2,7,2)

What if the gear jam were corrected? Well the values would get larger, but perhaps not by as much as you think. The answer would still not involve much more than a googol up-arrows. Consider a baggoogol with this corrected definition...

baggoogol = g(2,10^100,10^100+1,10^100,2) =

g(10^100,10^100+1,g(10^100+1,10^100,2),2)

In this case we would apply "^^^...^^^2^^^...^^^googol" to "2" a googol times! By using PAL we can find a simple upperbound for g(a,b,c,d). Just observe.

g(1,b,c,d) = g(b,c,d)

g(2,b,c,d) = g(b,c,g(b,c,d)) < g(b,c+c,d) = g(b,2c,d)

if... g(k,b,c,d) < g(b,kc,d)

then...

g(k+1,b,c,d) =g(k,b,c,g(b,c,d)) < g(b,kc,g(b,c,d) < g(b,kc+c,d) = g(b,(k+1)c,d)

therefore...

g(a,b,c,d) < g(b,ac,d)

Thus ... 

                    By taking the 3rd rule and replacing b with 1, c with b, d with c, and e with d, we get the 2nd rule. Thus the 2nd rule is redundant. What he have here is a simple primitive recursion. But a primitive recursion requires a base case. Joyce doesn't seem to realize that he needs to specify this, or he considers it "obvious". Lucky for us, however, it can easily be inferred from some of his later examples. For example Joyce defines:

googolduplex = g(4,1,2,2,10)

googoltriplex = g(5,1,2,2,10)

We already know that a googolduplex = 10^10^10^100 , and a googoltriplex = 10^10^10^10^100. Because we know what values Joyce is going for we can use this to help us interpret Joyce's expressions. Applying rule 3 we have...

googolduplex = g(4,1,2,2,10) = g(3,1,2,g(2,2,10),10) = g(3,1,2,100,10) =

g(2,1,2,g(2,100,10),10) = g(2,1,2,10^100,10) = g(1,1,2,g(2,10^100,10),10) =

g(1,1,2,10^10^100,10)

Now note, if we let 1 be the default value, and establish the rule:

g(1,@) = g(@)

Whenever g contains more than 3 arguments we can say...

g(1,1,2,10^10^100,10) = g(1,2,10^10^100,10) = g(2,10^10^100,10) = 10^10^10^100

which is the correct value for a googolduplex. So it works! We can also see what is happening pretty clearly here. The 5th argument allows us to iterate the polyponent. It therefore can be used to very good effect to define numbers that fall between the cracks of up-arrow notation, such as those in the googol family. We can also see that...

googoltriplex = g(5,1,2,2,10) = g(4,1,2,g(2,2,10),10) = g(4,1,2,100,10)

= g(3,1,2,g(2,100,10),10) = g(3,1,2,10^100,10) = g(2,1,2,g(2,10^100,10),10) =

g(2,1,2,10^10^100,10) = g(1,1,2,g(2,10^10^100,10),10) = g(1,1,2,10^10^10^100,10)

= g(1,2,10^10^10^100,10) = g(2,10^10^10^100,10) = 10^10^10^10^100

Furthermore we can see that...

googol-(n)-plex = g(n+2,1,2,2,10) = g(n+1,1,2,100,10) = g(n,1,2,googol,10)

I like to call googol-(99)-plex a "grangol" (formed from grand-googol). In this notation we have...

grangol = g(100,1,2,100,10)

As you can see, the 5-argument g-function is perfectly well defined, and is even functional. This still does not produce anything beyond the Ackermann function however! 

For example ...

g(n,1,2,100,10) < g(3,n+2,10) 

What about the 6-argument case? Is it well defined, and is so, is it more powerful than the Ackermann function?

Rules 4 and 5 deal with the 6-argument case, but again the 2 rules are redundant and can be reduced to a single case. In this case, the 4th rule suffices for cases involving 6-arguments...

g(a,b,c,d,e,f) = g(a-1,b,c,g(d,e,f),e,f)

Once again a base case is needed. Joyce doesn't have any examples of worked out 6-argument expressions, but we can gather that the rules are basically the same. First of all it's clear that rules 2 and 5 are using 1's as place holder values, suggesting that 1's are default for all arguments past 3. Thus 1 will be the terminating case for 6-arguments and beyond, and will simply be dropped in such a case. With this in mind, it turns out that finally Joyce produce's something which grows faster than the Ackermann function and is on par with Graham's function (we will learn about Graham's function in this chapter). To give an example of how this works consider the following...

g(1,1,1,4,3,3) = g(1,1,4,3,3) = g(1,4,3,3) = g(4,3,3) = 3^^^3 (HUGE!)

Now let's try a 2...

g(2,1,1,4,3,3) = g(1,1,1,g(4,3,3),3,3) = g(g(4,3,3),3,3) = g(3^^^3,3,3) =

3^^^^^^^^^ ... ... ^^^^^^^^^3

w/3^^^3-1 ^s

Woah! Where did that come from! The 6-argument iterates the "degree" of the Ackermann function. This leads immediately to an increment in what is known as the "order-type" (order-type is a more advanced topic we will learn more about in Section IV). This "increment" is the most basic building block of recursive functions. Yet Joyce has wasted 3 arguments to get here from the Ackermann function. No matter. All is forgiven, because we now actually have something to write home about. The next step is scary big...

g(3,1,1,4,3,3) = g(2,1,1,g(4,3,3),3,3) = g(1,1,1,g(g(4,3,3),3,3),3,3) =

g(g(g(4,3,3),3,3),3,3) =

3^^^^^^^^^ ... ... ... ^^^^^^^^^3

w/3^^^...^^^3-1 ^s

w/3^^^3-1 ^s

Each time we increment the 6th argument we feed back the number of arrows into itself! This leads to a growth rate here-to-for unencountered! In short, this beats up-arrow notation, because the number of arrows quickly becomes more than can be written out in the universe!

The final rule appears to be . . . patalogical? Note that the form does not follow any other expression in JMGEN. The arguments are not simply separated by commas, but two are grouped within paratheses. Does this mark some special status? This might be okay if what followed wasn't complete nonsense...

g((a,b),c,d) = g((a-1,b-1),g((a,b),c-1,d),d) = g(a-1,g(b-1,g(b,c-1,d)-1,d)

Let's take the first rule at face value. This is at least well formed. Now let's try and example...

g((3,3),3,3) = g((2,2),g((3,3),2,3),3) =

g((2,2),g((2,2),g((3,3),1,3),3),3)

At this point we run into a serious problem, because if we continue we are going to keep decrementing indefinitely. Problem is, we aren't told what the base case is! Even if we assume it's a 1, we aren't told what we are suppose to do here. The second statement is even worse and isn't even syntactically correct! Note that it is missing a closing paratheses...

g(a-1,g(b-1,g(b,c-1,d)-1,d)

                Like rule 1, rule 6 never comes into play, so it doesn't seem to make much of a difference.

                What about going beyond 6-arguments? Well the rules don't tell us what to do in this case. This would mean the best we could hope for is some clues from some worked out examples. Turns out however, that Joyce only makes very sparing use of the g-function past 6-arguments. So let's proceed with Joyce's googology for the time being. When we get to examples involving more than 6-arguments we will return to this issue...

More Joycian Googology

            Joyce continues his system, expanding on his "bar" root, in ways that aren't clear to me. He defines...

umpbarbar = g(1,g(2,1000,1000),1000) = 1000*1000^1000 = 10^3*10^3000 = 10^3003

(also umpdubar )

umpbarbarbar = g(1,g(2,g(2,1000,1000),1000),1000) = 1000*1000^1000^1000 =

10^3*10^(3*10^3000) = 10^(3*10^3000+3)

(also umptribar )

                For whatever reason the text has this bad habit of discussing other peoples work and generally getting it wrong in one way or another. There is little indication of whether this is part of Joyce's work or additional commentary on the part Michael Halm, presumably to "demonstrate" that Joyce's notation is already capable of matching or surpassing other notations. This leads to an epic (and I mean EPIC!) fail when we get up to the work of Jonathan Bowers.

                In the meantime here's what Michael/Joyce did to f--k-up the work of Alistair Cockburn (remember we talked about his work earlier in the chapter). Firstly the text states...

gargoogol = g(2,2,g(2,100,10)) = (10^100)^2 = 10^200

                This is actually correct. The problem. Earlier in the text it says that Kieran defined this as a googolplex googolplexes. That would be a gargoogolplex. A gargoogol would be a googol googol's of coarse. It get's worse however. Next it defines...    

fuga-(n) = g(3,2,n) = n^^2 = n^n

                    This is not correct if you go back and read Cockburn's original article. It's...

fz-(n) = g(3,2,n) = n^^2 = n^n

To Joyce's credit he does give the correct definition for megafuga- ...

megafuga-(n) = g(4,2,n) = n^^^2 = n^^n

(finally something correct)

The text also attempts to define the "gag" suffix using the g-function...

gag-(n) = g(n+1,n,2)

The gag-(n) function is defined as A(n,n) using the 2-argument Ackermann function. For technical reasons, the conversion between the 2-argument and the 3-argument Ackermann function is not quite this simple, though it serves as an approximation. The text goes on to say that a Tom Kreitzberg suggested the continuation...

g(a,1,b,c,d) = g(a-1,1,b,g(b,c,d),d)

g(a,b,c,d,e) = g(a-1,b,c,g(c,d,e),e)

Joyce's More Generalized Exponential Notation

                    Joyce's More Generalized Exponential Notation (referred to as JMGEN forth with) allows for more than 3 arguments in the g-function. The problem is that the rules are either, ambiguous, broken, or poorly designed. The following 6 rules are offered...

g(a,b,c,d) = g(a-1,b,c,g(a,b,c))

g(a,1,b,c,d) = g(a-1,1,b,g(b,c,d),d)

g(a,b,c,d,e) = g(a-1,b,c,g(c,d,e),e)

g(a,b,c,d,e,f) = g(a-1,b,c,g(d,e,f),e,f)

g(a,1,1,b,c,d) = g(a-1,1,1,g(b,c,d),c,d)

g((a,b),c,d) = g((a-1,b-1),g((a,b),c-1,d),d) = g(a-1,g(b-1,g(b,c-1,d)-1,d)

                    This is how the rules appear verbatim in the text. To the untrained eye, these may appear indistinguishable from any other googological notation, as a random assemblage of mathematical statements that define some function. However, not all such assemblages produce large numbers, and one also has to be careful to make sure it is well-defined and doesn't contain infinite loops. We see that the 1st rule deals with 4 arguments, the 2nd and 3rd rule deal with 5 arguments, the 4th and 5th deal with 6 arguments, and the 6th rule, as far as I can tell is pure nonsense (it might be patalogical). Earlier we got a rule for the 4th argument g-function from the example with a googolplex, but this is inconsistent with the rule for 4 arguments presented here.

                    For convenience, I'll reverse the order of the arguments. The first argument, will be the furthest right, and denoted by a, the 2nd argument will be the next from the right, denoted by b, and so on. Here is the previously implied rule...

g(1,c,b,a) = g(c,b,a) 

g(d,c,b,a) = g(c,g(d-1,c,b,a),a)

As noted earlier, this only produces a redundant recursion no stronger than the recursions found with 3 arguments. But it get's worse because the new rule is nothing like the above. Let's work out the example given earlier for a googolplex , g(3,2,2,10) , and see how it evaluates given the new rule:

g(3,2,2,10) = g(2,2,2,g(3,2,2)) = g(2,2,2,2^^2) = g(2,2,2,4) = g(1,2,2,g(2,2,2))

= g(1,2,2,2^2) = g(1,2,2,4)

Here we need to make a leap, because Joyce has failed to include terminating conditions. Since "1"s are used as place holders in the other rules, it seems that Joyce intends to use "1"s as terminating values for arguments past 3. In that case we have...

g(1,2,2,4) = g(2,2,4) = 4^2 = 16

So instead of getting a googolplex, we get a measly 16! But perhaps this is just a fluke. What if we change the last argument from a 3 to a 4? Well then we get...

g(4,2,2,10) = g(3,2,2,g(4,2,2)) = g(3,2,2,2^^^2) = g(3,2,2,4) = g(2,2,2,g(3,2,2))

= g(2,2,2,2^^2) = g(2,2,2,4) = g(1,2,2,g(2,2,2)) = g(1,2,2,2^2) = g(1,2,2,4) = g(2,2,4)

= 4^2 = 16

16 again. Hmm. Maybe this is just a degenerate case, due to the 2's. So let's replace b with 10 and c with 3 and try the sequence d=1,2,3,etc. That will probably give us a pretty good idea what is going on with this recursion ...

g(1,3,10,10) = g(3,10,10) = 10^^10 = HUGE!

... good start ...

g(2,3,10,10) = g(1,3,10,g(2,3,10)) = g(3,10,10^3) = g(3,10,1000)

= 1000^^10

This is a rather strange result. Even though it's technically larger, tetration is a very powerful operation and has counter-intuitive properties. Having a large base like 1000 doesn't make much of a difference in this case. This number will still be vastly smaller than 10^^11. Not a good sign for a recursion. Let's see what happens next...

g(3,3,10,10) = g(2,3,10,g(3,3,10)) = g(2,3,10,10^^3) = g(1,3,10,g(2,3,10))

= g(3,10,g(2,3,10)) = 1000^^10

what?! This is no larger than before. What happens with d=4?! ...

g(4,3,10,10) = g(3,3,10,g(4,3,10)) = g(3,3,10,10^^^3) = g(2,3,10,g(3,3,10))

= g(2,3,10,10^^3) = g(1,3,10,g(2,3,10)) = g(3,10,10^3) = 1000^^10

It appears the rotor is stuck! I'll call this being gear jammed. Although larger and larger values keep cropping up in the evaluation process they keep getting overwritten until we reach the base case of a mere 1000^^10. Even if the overwrite wasn't occurring, we still are only going up an arrow at a time, which we could have just achieved by incrementing c in the 3-argument version! It appears that Joyce has carelessly slapped a definition together without considering whether it actually produces large values! As a general rule of thumb, it should not be assumed that any set of transformations necessarily leads to larger and larger numbers.

Some further analysis reveals the problem. In the rule...

g(a,b,c,d) = g(a-1,b,c,g(a,b,c))

notice that the substitution occurs in the spot where "d" was (ie. the 1st place). Yet by the same token g(a,b,c) does not take d as an argument. The result is that d keeps being overwritten with smaller and smaller values. This occurs until we reach the base case. This can be seen by applying the formula an arbitrary number of times...

g(a,b,c,d) = g(a-1,b,c,g(a,b,c)) = g(a-2,b,c,g(a-1,b,c)) = g(a-3,b,c,g(a-2,b,c)) =

... = g(1,b,c,g(2,b,c)) = g(b,c,g(2,b,c))

Thus ...

g(a,b,c,d) = g(b,c,g(2,b,c)) = g(b,c,c^b)

                    Notice that it doesn't matter what values "a" and "d" are. Did Joyce intend something different, or was he simply coming up with random rules for the pataphysics of it. I don't know. What I will say is this is why Pataphysics will rot your brain! Ironically, we who call ourselves googologist's have moved beyond the purely pataphysical, preferring instead the sensible and concrete abstraction rather than the purely nonsensical one.

                        At this point all we can do is speculate. What if a typo lead to this definition when what was actually meant was ...

g(a,b,c,d) = g(a-1,b,c,g(b,c,d))

This actually makes a lot more recursive sense, although it's still a little odd. Here the counter (a) isn't being copied but stored in the 4th argument. However we run into another problem. Let's go back to our previous example...

g(1,3,10,10) = g(3,10,10) = 10^^10

g(2,3,10,10) = g(1,3,10,g(3,10,10)) = g(3,10,10^^10) = (10^^10)^^10 ~ 10^^20

That's way better than before ...

g(3,3,10,10) = g(2,3,10,g(3,10,10)) = g(2,3,10,10^^10) = g(1,3,10,g(3,10,10^^10))

= g(3,10,g(3,10,10^^10)) = ((10^^10)^^10)^^10 ~ 10^^30

So what we have here is a genuine iteration now. The problem is, it's iterating in the base instead of the polyponent. The result is again, grow rates already covered by the 3-argument function. For a googologist, Joyce appears to be no better at it than the average modern beginner! Googology is not just slapping rules together in a haphazard manner. It is a skill that has to be learned and developed for the purpose of coming up with the largest numbers possible.

                    It turns out however to make little difference since Joyce never actually use the 4-argument g-function, except in places where it is clearly in error (as pointed out earlier).

                    More interesting is the 5 argument version. Rules 2 and 3 deal with the 5-argument version. It turns out 2 rules aren't necessary and a single rule covers both cases.

xova = g(2,27,10) = 10^27

weica = g(2,30,10) = 10^30

vunda = g(2,33,10) = 10^33

uda = g(2,36,10) = 10^36

treda = g(2,39,10) = 10^39

satta = g(2,42,10) = 10^42

rinda = g(2,45,10) = 10^45

qeda = g(2,48,10) = 10^48

pica = g(2,51,10) = 10^51

oca = g(2,54,10) = 10^54

nica = g(2,57,10) = 10^57

menta = g(2,60,10) = 10^60

                    Beyond this point the text becomes rather cryptic. Here is the excerpt for how to continue ...

   Beyond menta- -(e)nta- is used until sara- [s + quaranta] g(2, 120, 10), -(a)ra- until inqua-  [i + cinquanta] g(2, 150, 10), -(i)nqua- until yessa- [y + sessanta] g(2, 180, 10), -(e)ssa- until otta- [o  + settanta] g(2, 210, 10), (e)tta- to fetta- [f + settanta] g(2, 237, 10). To prevent confusion with eta- or yotta- higher infixes would have increase to trisyllablic, (o)ttanta- until uvanta- [u +  novanta] g(2, 270, 10), (o)vanta- to lovanta- [l + novanta] g(2, 297, 10).  This is the limit of this system since to apply it any higher we would have both kenta- [k + venti] g(2, 66, 10) and [k + cento] g(2, 300, 10).

--Joyce [sic]

                    As it so happens Joyce makes little use of all this, except for lovanta = g(2,297,10), which forms the basis of one of his googolism's.

                    To go much further in Joyce's work we now have to face it's most perplexing feature ... a rather ad hoc multi-argument extension of the Ackermann function.

He goes on in a style inspired by the works of Aronson and Blowers, by going backwards through the alphabet while reciting latin with slight modifications along the way. Here is his list of extended prefixes...

mega = g(2,6,10) = 10^6

giga = g(2,9,10) = 10^9

tera = g(2,12,10) = 10^12

peta = g(2,15,10) = 10^15

exa = g(2,18,10) = 10^18

zetta = g(2,21,10) = 10^21

yotta = g(2,24,10) = 10^24