# Introduction to Googology

**So you wanna be a googologist?**

So you wanna be a googologist, but know little to nothing about it? Well you've come to the right place. I'll guide you through the basics, point out a bunch of common misunderstandings and rookie mistakes to avoid, and give you an overview of the subject and provide some resources. I can't promise you'll be an expert after reading this, but hopefully you'll have a better idea where to begin and where to look to learn more.

Before we get started here is some preliminary information...

**Preliminaries**

**What is googology?**

Let's begin with a definition. What is googology? A common answer is the googology is the "study of large numbers". However, this definition is somewhat apocryphal and misleading. How exactly do we *study large numbers*? A more accurate description is...

*Googology *is the practice/craft of generating, comparing, and naming large numbers and the study and theory of how to do so.

Googology is not JUST a study. It is a craft. A craft is something which is more than merely theoretical and is focused on producing an actual end-product. In the case of googology, we *craft *"googolism's", names for specific very large numbers. It is also not so much a study of large numbers, as a study of how to *create *large numbers. The numbers themselves are usually of no particular interest except for their size and how they compare to other numbers.

**Objective**

What is the objective of googology? The objective is to "name" as large a number as possible, or alternatively to perpetually find better and better ways to "name" larger and larger numbers. The secondary objective of googology is to name as many numbers as possible along the way to larger and larger numbers.

**Googology Community**

The googology community is the group of individuals who are part of this wiki, and related wiki's, as well as non-members who participate in googology. As a new member of this wiki you are part of the googology community, but you are not yet a googologist.

**How to Become a Googologist**

To become a googologist you must contribute something * new *to googology. Since our subject is literally boundless, there is always a need for new ideas. That being said, before you dive right in, let me give some pointers to avoid the common pitfalls for beginners. Here is a list (from most significant to least) of do's and don'ts for all fledging googologist's ...

**THE 4 COMMANDMENTS OF GOOGOLOGY **

**(MORE SOON!)**

**#1 - DON'T USE THE WIKI'S DATABASE TO INTRODUCE YOUR NUMBER! **

**Create your own publishing Platform instead**

This is a very common mistake that beginners make. Perhaps you have, or about to do the same. It's happened many times before. Someone is new to googology, has some great new idea for a number, creates an article solely to introduce their number and ... the article is promptly deleted, with the only justification being "no sources cited"?

*No Sources! Why would a source be needed? Isn't this a website about totally made up stuff!*

It's true that almost nothing in googology is "officially" recognized outside of googology, with only a few rare exceptions (ex. *googolplex, Graham's Number, etc.*). But so what. The googology community considers numbers represented on this wiki as part of established googology canon. Inevitably, googology wiki (and it's foreign language counterparts) serve as an official record and standards bearer for the googology community simply because this is as "official" as it gets for us. If you were attempting to "sneak" in your number by publishing an uncited, uncredited article, then you yourself have already recognized this and were trying to make your number "official". But wait, you say, if it's posted on the wiki then it's de facto "official" because it's now part of googology wiki, which officiates it! Try to use that reasoning to justify an uncited article in wikipedia and see if it flies. It doesn't fly here either.

Firstly, let's consider what it means for something to be "official" anyway. The reason a number like a "googolplex" is recognized in the english dictionary is simply because it is wide-spread enough to merit inclusion in their dictionary. In other words, dictionaries have *standards *that have to be met for inclusion. Believe it or not googology wiki has standards of inclusion as well. However the bar has been set to **THE ABSOLUTE BARE MINIMUM! Currently only a single outside source is required to justify the existence of any googolism as offical!**

That's right. The process of making your googolism "official" has been made as easy as possible. Googologist's don't care so much about whether a googolism is in wide use ... almost by definition no googolism would ever be in wide use because they are TOO BIG TO BE WIDELY USEFUL. Googologist's also don't care whether your a Carnera or an Einstein ... as long as your googolism is a well defined number with a name, it counts! That's really almost too generous. So really you are in the best possible position to have your googolism heard here at googology wiki.

*Okay then, so why can't the "article" itself serve as the source? *

Well, that would be like an editor of a dictionary using his own entry in the dictionary to justify its use! It's circular. A dictionary, or an encyclopedia, does not invent, it catalogues things which already exist outside itself. Therefore the Wiki's database should never be used as a publishing platform.

Don't use the Wiki's Database to introduce your Number. It doesn't look good. For as long as this wiki has been around people have tried to vandalize it, pull hoaxes, and generally toss in anything that comes to mind. It doesn't look constructive when you do this. As a member of the googology wiki, you should be helping to gather information on googology on the web, not using the wiki as your person platform.

*"this is just a conspiracy by googologist's to keep non-googolist's from inventing numbers".*

This is not true. No one is stopping you from inventing any number you want and calling it whatever you want. But you are seeking *recognition from others *for your work. Therefore you must meet the bare minimum standard for entry. As long as your number is well-defined, there is really only one major reason your "number" wouldn't be included, and that would be for lack of an outside source. The only other reasons it might be excluded are because it was ill-defined, or was not sufficiently notable (we'll look into these other considerations later). Googology Wiki doesn't just include large numbers from professional mathematics, like *Graham's Number. *It includes things like "*zootzootplex*" , which was coined by a 4 year old. So it is not for lack of sophistication that your number was excluded, it was for lack of an outside source. The general philosophy of this wiki has been to be as inclusive and total as possible. A single source ... that you will create yourself anyway, is not too much to ask.

*But why? What's the difference if I create it here or somewhere else?*

Think of it from the wiki's point of view. If people can simply post whatever they want, whenever they want, then the wiki becomes a free-for-all. Eventually we would be flooded by endless articles like "googolplex^1000^1000+10^(mega!!)", and they would quickly outnumber the useful articles about a "googolplex" and "mega". The fact of the matter is, that the wiki can not include every number conceivable, even between 0 and a googolplex, because there are simply too many. So some selection is still required even at googology wiki. But as long as your number doesn't take 15 pages to describe a number less than a googolplex ... it will probably get included ... as long as you have a source.

*So how do I get a source? *

There are plenty of other venues for publishing your number. In fact, these days it's easier than ever before. First and foremost, if you have a user account here, you can create your own blog entries. So go ahead and create a blog about your numbers. You can also just use your own user pages, like your profile or talk page. What else can you do? Did you know that google offers a free website service to anyone with a gmail account? Seriously, check it out. You could create a website to host your numbers on, and then you'll meet the source requirement. Another option is a service called wix, which also let's you create a free website. If your serious about becoming a googologist then step#1 is CREATE A PUBLISHING PLATFORM FOR YOURSELF. Preferably it should be a website.

**#2 - NEVER USE INFINITY IN YOUR NUMBER, EVER!**

*So googology is about coming up with the biggest number huh? Well guys I don't know why you didn't think of this before, but your loss ... my number is Infinity! Do I have the biggest number now?*

This is another one of the most common rookie mistakes. Don't use *infinity *in your number, ever. To the common person using *infinity *seems like the obvious thing to do, but in googology no one is going to take you seriously if you do. Do you really think googologist's don't know about infinity? Our very subject depends on the fact that there are indeed an infinite number of finite numbers. So *infinity *is just as well known in googology as the number *zero, *both are given at the outset in the definition of numbers!

If your going to be a googologist, your going to need to broaden your perspective on infinity. For the average person ... *infinity is a number, it's the biggest number. *Let me give you an idea what *infinity is* to a googologist.

Imagine the following function. Let f(1) = 2, and let f(n) = f(f(n-1))+1. Now what is f(2). Well using the rule we get the following...

f(2) = f(f(1))+1 = f(2)+1 = f(f(1))+1+1 = f(2)+2 = f(f(1))+1+2 = f(2)+3 = f(f(1))+1+3 = f(2)+4 = f(2)+5 = f(2)+6 = f(2)+7 = f(2)+8 = ...

Now ask yourself ... what value does this result in. *Infinity*, right? **WRONG! **It doesn't have a value! In googology the value of an expression is the number that results at the *end *of an evaluation process. For example 6! is a number because it can be evaluated ...

6! = 6*5! = 6*5*4! = 6*5*4*3! = 6*5*4*3*2! = 6*5*4*3*2*1! = 6*5*4*3*2*1 = 6*5*4*3*2 = 6*5*4*6 = 6*5*24 = 6*120 = 720.

So 6! is just another "name" for 720. Mind you, in many cases the initial expression will take too long to completely evaluate for us to obtain it's full decimal expansion. For example, 6!!!. In googology however, this doesn't matter. We know that 6!!! will eventually complete it's evaluation and result in some truly massive value. So 6!!! is a number (it's bigger than a googolplex in fact!). But f(2) NEVER finishes evaluating and so never produces a value ... therefore it's not infinity ... it has no value. Come to think of it, what IS the value of infinity. What *number *is infinity. Well there are an infinite number of numbers, so if we try to count all the numbers, that *count *represents infinity. So let's say there IS a number that represents infinity. But then you could find infinity among the numbers and it would have a successor. But then infinity would not represent all the numbers and would represent a subset of them. Therefore infinity can't have a number. It is literally innumerable, not just in practice but in principle!

Now apply this to any concept you would normally attach to large numbers and by extension to infinity, and nullify it. So infinity isn't big, it's big-less, it's not vast, it's vast-less, it's not the largest number ... it is number-less ...

So to a googologist *infinity *is not the overabundance of number ... it's the complete absense of it! In fact, in the Busy Beaver game, turing machines which never end ... don't get a score of infinity ... they get a score of -1. I suggest we adopt this standard ...

const int INFINITY = -1;

In other words, in googology, you can define infinity as the same as -1. It's a score even lower than 0, the lowest number in googology! Furthermore, any expression which contains infinity will also be infinity (read -1), so never ever use it because it would be worse than making 0 your number.

We've seen it all before here at googology wiki ... 10^infinity, 10^(3*infinity+3), infinity!, 2^infinity, infinity^infinity, infinity^^infinity, infinity-->infinity-->infinity-->infinity, etc.

As "values" all this amounts to pure nonsense in googology. Mind you, there are useful ways to use infinity in googology, and there are theories for how to work with infinities, but these must never be treated as "googolism's". Remember ... googology wiki only excepts finite values in the "Numbers" category.

**#3 - MAKE SURE YOUR NUMBER IS A DEFINITE AND WELL DEFINED VALUE**

*Alright guys, I've got you all beat this time! My number is the largest number possible. It's 1/infinitesimal. It's larger than Graham's Number, or even Rayo's Number, because the infinitesimal is smaller than any positive real number. In fact it's bigger than any finite number. But it's smaller than infinity because the infinitesimal is greater than 0, and everyone knows 1/0 = infinity. So I have defined the largest number!*

This is another really common mistake, and it actually takes on a whole host of forms. I won't be able to go through all of the kinds of mistakes of this variety, but I'll try to point out the more common kinds of mistakes. First and foremost, don't try to "cheat" and come up with some round about way to come up with a "largest number". That is just as bad as trying to use infinity. Basically, if ever your strategy involves some way that there is no possible come-back ... then it's not legit ... PERIOD! I consider these "game-breakers". Game-breakers don't count, they aren't numbers, they are ill-defined. In googology your goal is to come up with an honest-to-goodness real ACTUAL number that you can point to and say exists. The catch is that *there is always a larger number someone else could come up with, even if it just means adding 1. (What's the point then? We'll address that later)*

At the outset googologist's take it as axiomatic that there is no largest number, where I use number to mean non-negative integer, ie. a member of the set {0,1,2,3,...}. This set doesn't have a largest member, so if you claim your number some how doesn't have a successor, then guess what ... it's not a member of the set and it doesn't count, NO EXCEPTIONS. Sometimes people try to act like some sort of authority, they may try to use complicated arguments to justify that there IS a largest number. Don't be fooled. Any child should no better. You don't need to be a professor to know there is no largest number ... it's axiomatic!

I suppose for most of you that goes without saying, but there are plenty of other ways to have not well-defined numbers. For one, don't attempt to use undefined expressions as your "number" like 1/0, -log0 , or other equally ridiculous expressions.

Okay, now that we got that out of the way, let's get to the more subtle cases. Fledging googologist's have a tendancy to be over-eager and want to jump into the deep end. Once you realize how pitiful addition, multiplication, and exponentation is here, the next most likely thing you are to do is grab a bunch of powerful notations you barely understand and smoosh them together in some haphazard manner. While this isn't always going to blow up in your face, and this will sometimes produce an actual number, carelessly jumbling together notations that you don't understand can result in meaningless or non-terminating expressions. Each notation has very precise syntax, and if you don't follow the syntax your expression has no meaning. As a rule of thumb don't carelessly manipulate notations you don't have any understanding of. Don't just assume that ANY EXPRESSION is automatically meaningful and representative of some number. Remember:

NO NUMERIC NOTATION IS INHERENTLY MEANINGFUL. THE MEANING MUST ALWAYS BE DEFINED BY THE HUMAN OPERATOR

Think of it like this. A sophisticated notation is like a over-complicated control scheme for a jet airliner. Would you want to fiddle with the controls with no experience? Well the same applies in googology. If you don't understand how a notation works you might create a broken expression that doesn't actually produce a number. Now mind you, you don't have to fully understand a notation to actually use it, or to produce a number. Take any expression that has numeric arguments. Those arguments can usually be replaced by any positive integer, or any mathematical expression which evaluates to a positive integer. In most cases, if you make sure that the values you are replacing are larger than the original values, and you are not reducing any of the values, then the new expression would be valued higher. Keep in mind though that functions have domains and you have to make sure the numbers you use are in the domain. For example, you can't plug decimal numbers into a array notation ...

{5,4,1.41}

No one has yet defined real-valued up-arrows, so this expression doesn't have any meaning until YOU define it. Don't expect it to be a given that this *evaluates *to something, and don't expect others to guess what you mean. In order for your number to be well-defined YOU MUST DEFINE YOUR NUMBER EXPLICITLY.

So even dealing with the numbers in an expression can have stags. However you will generally be safe as long as you only use positive integers, because most googological notations are defined for the positive integers (and sometimes 0). Be extra precautious however when dealing with the non-numeric characters in an expression. Unlike digits, which can be strung in any arrangement to produce a number, the non-numeric characters can only be arranged in certain ways to define well-formed expressions. Even if you do manage to place them in a combination that makes sense without an understanding of the notation you won't know which non-numeric strings are more powerful than others. The results may be that your number is only *slightly larger*, or worse ... that it is actually smaller! As a general rule of thumb, don't just use a large number notation, try to understand how it actually works. Otherwise, odds are, you won't be doing anything very impressive with it.

Here's another important point. Sometimes people will take this tack ...

*my number is so amazingly big ... you just can't even imagine how big it is ... it makes Rayo's Number look absolutely insignificant. It's an indescribable number ... it would give God a headache ... blah blah blah*

Don't expect googologist's to be impressed. Just show us the money. Define your number and we shall see what it does and does not dwarf, otherwise it's just hogwash.

Which brings me to another point ... googologist's are allergic to the indefinite. We are only interested in definite numbers. Stuff like "a lot" , "many" , "a huge amount", mean nothing to a googologist, in and of themselves. These are simply vague-pseudo-quantifiers. They should only be used informally, never as a definition. This goes also for things which sound like numbers, but which have no formal definition, such as a *zillion, jillion, gazillion, infinitillion, *etc. If it's just a word made to sound like a googolism, but it has no definition ... it's meaningless.

Don't try being clever, by using meaningless circular references.

Some examples are, self-reference ...

*unfathomillion *= 10^*unfathomillion*

In order to know what unfathomillion is, we first have to know what unfathomillion is! Sometimes self-reference may actually define a value. For example

x = (1.444667810098...)^x

In this case, x = 2.718281828..., or 'e'. This is what is known as a "fixed-point". It's a point at which the output is equal to the input of a given function. However the function f(x) = 10^x has no fixed-points! So *unfathomillion *is meaningless!

*What about infinity? doesn't infinity = 10^infinity*

Yeah but infinity isn't part of the functions domain, and even if I were to except something like that ... infinity is still useless as an answer! We need a finite number!

Next are more complex chains of references ... which always fail to actually define anything. For example

*gothouzillionogol *is defined as 37 *kedrillion*

and guess what...

a *kedrillion *is 856 *gothouzillionogol*

This is just as useless as before. It would also be possible to create very long circular chains. But if at any point in the chain a number refers to a number earlier in the chain, then what you have is just a very elaborate self-reference. It's meaningless! Don't do it.

Don't attempt to create infinite strings of digits. In all cases this will have no meaning. Some examples...

100000.... where there are an infinite number of zeroes (then the place value of 1 is undefined!)

314159265359.... (pi with the decimal removed. Same problem, undefined place values. Also impossible to evaluate the modulus ... which is nonsense!)

2718281828... ('e' with the decimal removed)

Ditto for putting these in reverse ...954562951413 and ...8281828172.

What else could go wrong? Well besides just flat out being ill-defined there are more subtle ways to not have a defined value.

You should probably study how other notations work before creating your own, but if you do decide to create your own please be aware of these common pitfalls.

1.** Don't rely too heavily on a case by case basis.** You should be working towards creating a rule-set which defines what has to be done next for any particular case. If you go with a case-by-case basis, instead of trying to generalize and understand your system, there is a higher probability of making mistakes and leaving things either ambiguous or undefined. Always try to compact your description of your notation as much as possible.

2.** Don't assume that some vague sequence HAS to have a defined continuation**. Sometimes people make this mistake when they are trying to "break out" of the system. An example might be something like this...

f(1) = 1

f(2) = 2+2 = 4

f(3) = 3^3 = 27

f(4) = 4^^4

f(5) = 5^^^^^5

f(6) = {6,6,6}

f(7) = 7-->7-->7-->7

f(8) = {8,8,8,8,8,8,8,8}

...

This will be advertised as some function which keeps growing faster than itself ... somehow. There is usually an assurance that f(100) or f(1000) is probably much much bigger than a *meameamealokkapoowa oompa, *or even *Rayo's Number. *Problem is, that the person who came up with sequence has no idea how it works or how it continues. So they will usually ask an experienced googologist "what comes next?". Well, nothing. You haven't defined a rule that determines what the next member of the sequence is. Your just haphazardly creating a sequence an item at a time hoping to outrun your own meager recursion. Such a sequence is necessarily finite. In order to have an infinite sequence you need a rule which generates the nth term. Outside of that, you can define a finite sequence an item at a time, but you'll always have to stop somewhere. And if your just going to declare the 100th or 1000th member bigger than a meameamealokkapoowa oompa anyway, why not ditch the whole ruse all together and just define your number directly. Defining an actual sequence is hard work, especially if you want it to be an extremely fast growing one. That is why this sort of "sequence" is appealing because it seems to side-step that somehow by jumping from one sequence to another. But don't be fooled. This is a mirage. If you want to know what your actually looking at, it's the concept of diagonalization ... except that your avoiding commiting to an actual diagonalization because you know it will be "too small". I won't go into details here, check out part II - Nuts & Bolts for that. For now, just keep in mind that unless you have a definite rule, a sequence containing a few terms doesn't automatically have a *natural *continuation. Don't expect googologist's to be impressed by such vague sequences.

3. **Make sure your notation will actually terminate.** This is a very easy mistake to make for beginners because usually it seems like other large number notations never end anyway, and also, because it seems like they only get longer and longer, not shorter, so if your notation keeps getting longer it doesn't mean it will never end. Well guess what, it doesn't gaurantee it won't either. Large Number notations are tricky ... they try to delay termination as long as possible ... but they can't go too far or else they "go infinite", which is ALWAYS a bad thing in googology. The trick is subtle, and I can't explain it in detail here, but it essentially involves some hidden mechanic that IS getting smaller even while everything else seems to be blowing up! Think of it like this. Hidden within every large number notation is a clock ... which ticks slower and slower. When you write an expression in that notation you set the clocks timer to some finite (though possibly massive) number. Each tick takes longer and longer to pass, yet the notation gaurantees that the time between each tick is always finite. Since there are a finite number of ticks, each a finite amount of time apart, no matter how many ticks there are or how far apart they are the clock eventually terminates! The mistake that beginners make is that every time the clock ticks ... it keeps showing exactly the same time. The result is an infinite loop and the inability for the notation to terminate. As general advise, don't jump right into the most advanced kind of recursions when you are just learning about the simplest kinds. It's very easy to write something which is broken or non-terminating. Googology is not something your going to pick up and master during your lunch break. It will take time to master the new concepts.

4. **Don't grab disseparate concepts from various notations and just assume that they click together**. If they have never been put together before, then most likely there is some work that needs to be done first. This may involve a careful reworking of the concepts of one system to fit it into the other system. Think of it this way. Do you think someone with no experience could just hook up whole components from a jet airliner and a car and hope to build something functioning? Same applies here. Googological notations are like number-crunching engines, and simply colliding those engines together is not likely to produce a new functioning engine. Instead, you should tinker and take apart each engine slowly, understand how each component works, and you'll find you don't have to borrow components ... you could build the entire engine yourself from scratch!

One last thing before we move on to the next major item. Don't try to use the old trick "the largest number that can be defined in one hundred words." This leads to something known as the Berry Paradox, and even though it would seem that if we are choosing the largest defined number from a set of numbers we have a well-defined number we have a contradiction here. For you see, we just defined the number that should take 100 words ... in just 10. Now it is possible to define a larger number with less than 100 words. For example "two times the largest number that can be defined in one hundred words". And there you have it ... our old friend the circular reference. For you see we now have the radical claim that this is a number which is *larger than itself*. Which is nonsense. Whenever you want to find the largest number in a set of expressions, some of which define numbers, you must always ensure that you have a well formed language which can not make reference to itself. Don't try things like

"The largest number namable with a *googol characters in mathematics"*

unless your prepared to define what you mean by *mathematics*. However, no matter what definition you eventually come up with, it will not embody the whole of mathematics because your system S, must have a function S(n) for the largest value describable in S with n symbols, but this function can not exist in S if S is to form such a well-defined sequence. But then such a sequence lies outside your system, but it's still part of mathematics! So your system must necessarily be a sub-system of mathematics. Moral of the story : you can't contain all of mathematics in a bottle ... it is fundamentally boundless. If you want to create a well-defined value don't try to envoke the "ultimate system" to create it. There is no such thing as an ultimate mathematical system. Mathematics is simply the sum total of all bounded systems.

There is one last one I'd like to point out:

"The largest number humanity will invent by the year 1,000,000,000 C.E."

If you claim that THIS is your number ... then you've just triggered the Barry Paradox again. Because you see, then this number has already been invented in the year 2014 C.E. Damned circular references!

**#4 - NEVER USE AN OPEN ENDED OUTSIDE REFERENCE IN YOUR NUMBER**

**If you must reference something always be explicit about everything your actually including**

*So it can't be infinity, and it has to be well-defined, huh? Alright, I've got you guys this time. My number is the sum of all numbers on this wiki plus 1! Now I have the largest number! I'll call it [Insert Username]'s Number. Bet you guys didn't see that one coming*

* *We saw it coming ... from a hundred miles away -_-

This is one of the oldest and lamest tricks in the book. Probably since the dawn of large number competitions, there was always been some snarky person who has tried some variation of "my number is whatever your number is plus one". It's precisely these sort of school yard "tactics" which have kept googology from making further progress amongst the masses. If your one of the types prone to this sort of come-back then you probably have never bothered to think much about how far numbers go because ... why bother when you can just add one to someone else's. Let them do all the thinking, and you can take all the credit. But don't expect anyone to be fooled. No one is going to be impressed that ... YOU KNOW HOW TO ADD!

The first reason why this tactic is discouraged ... even if we were to agree it referenced a well-defined value ... you are still violating the 0th principle of googology ... your not contributing anything * NEW *to the discussion. You have simply taken the numbers in existence and combined them with addition. The point of googology is to invent better and better tools to create larger and larger numbers ... but your simply using one of the most primitive tools in the tool-set ... addition.

This is what I like to call a * vapid form*. The basic tactic of a

*is to take everything else in existence and combine it in some simple way that guarantees it's larger than any of it's constituent parts. The vapid form however does very little to contribute to the overall size of the resulting number ... the strength comes almost entirely from whatever is included within it. In this sense the vapid form can only be*

**vapid form***just as powerful as the whole of googology ... but*it itself contributes no new insights into it.

*But it is the largest number right? And it is well-defined, since the sum of any set of numbers is also well-defined. So what is the problem?*

There are a few problems with this. The first one is related again to the Berry paradox. Let's say (*god forbid!) , *that you post this monsterousity on the wiki. The article entry simply reads :

[Insert Username]'s Number is equal to the sum total of every number on this wiki plus 1

But now your "number" is a number in this wiki ... and we get a self-reference.

If your number is not on the wiki though this isn't a problem. The problem then is a little more subtle. The problem actually is ... that you haven't defined a number at all. WHAT? THAT CAN'T BE TRUE!

Well remember that googologist's are interested in defining definite exact values that can be compared to other values. What you have defined however, is not so much a number as a function which takes all the "numbers" on this site and adds them together and adds 1. It's just a polyadic function that could be defined as:

p(n) = n+1

p(@,n) = p(@)+n

where p() is applied to the whole of the googology wiki. Therefore you haven't actually defined a number, because the result depends on the precise content of the wiki. The problem is that the wiki is not a static entity. It changes from day to day. Numbers are added and deleted. There is nothing in the "definition" that accounts for this, so in theory the value would change from day-to-day. But then you haven't defined an integer constant, just a variable expression.

*Well alright, no problem then, I'll just define it as the sum as of right this MOMENT! *

Hmm ... that would define an integer constant wouldn't it. So I guess by next week your number will be hold hat. You see you won't be able to claim your number is larger than the largest number on the wiki by next week, because by next week a way larger number was added. But since your number only included the list of numbers on the wiki at that moment, a week ago, your number can not include it, and it is therefore smaller. Don't even bother with the fact that your number is the sum of all the old numbers ... it won't make a difference. The distance between googologically large numbers is such that the sum of every smaller googolism than G, will still be way way smaller than G. But's that's a problem isn't it v_v ... because you wanted a number that NO ONE could beat EVER! That's a game breaker. You can't have it both ways. Either you commit to a specific value and your value is subject to defeat, or you use an open-ended reference and don't define a definite value (so it's not a meaningful number entry). And that's pretty much why people try this. With a tactic like this you don't have to actually understand anything, but you don't contribute anything either.

Basically the problem here involves **outside references. **

**IN ORDER TO HAVE AN ACTUAL NUMBER YOU MUST PROVIDE A STATIC DEFINITION FOR THAT NUMBER, NO EXCEPTIONS**

Ideally your number's definition should be almost completely self-contained. We can take it for granted that the elementary functions are common knowledge and don't need to be explicitly defined. For the purposes of googology we can also assume that there is a common language that we write recursive functions in, which is composed of a bunch of formal transformations triggered by various conditions based on the current form of the expression. So a number like:

1000^1000^1000

is self contained. It doesn't require you to look up any other sources to figure it out.

On the other hand if we use...

G^G^G

where G is *Graham's Number, *then your definition is not self contained. In order to get the complete definition you must look up the source for "G" and * include it *within your definition. ie. you need something like:

#include "Graham.h"

define N := G^G^G;

When we include G into the definition we get the true definition which is...

return int {a,b,c} : {

if c=1 --> {a,b,1} = a^b ;

else

if b = 1 --> {a,1,c} = a ;

else

{a,b,c} = {a,{a,b-1,c},c-1} ;

}

return int G(n) : {

if n=1 --> G(1) = {3,3,4} ;

else

G(n) = {3,3,G(n-1) };

}

define G := G(64) ;

define N := G^G^G ;

... and this is the actual definition of your number. This defines an actual value that never changes. Your definition relies on the definition of "G" and G is defined explicitly based on exponentiation and the positive integers 3,4, and 64. If there is any ambiguity as to which Graham's Number you mean (there are actually 3 varieties), then guess what ... you have an ambiguous expression. So it may be better to just include the definition of Graham's Number your using, since it's not even that long anyway.

Now things can get quite complicated. You might include several outside sources to your definition. For example ...

#include "googolplex.h"

- include "mega.h"

define N := googolplex^1000^1000+10^(mega!!)

Furthermore those included sources may themselves require outside sources ...

//googolplex.h

#include "googol.h"

define googolplex := 10^googol

but eventually if we continue through any path of inclusions we must always reach a root source which does not require an outside source...

//googol.h

define googol := 10^100;

This is why circular references are not allowed. Because they would make it impossible to reach a root source.

Now you can include as many sources as you want ... as long as you are explicit about each and every source and where you got it from. The above tactic however would be the equivalent of ...

#include "googology.h"

define N := googology+1

where googology.h somehow includes all the sources on googology wiki, then defines their sum as "googology". But let's go to the source file of googology.h ...

//googology.h

#include "googol.h"

#include "googolplex.h"

#include "mega.h"

#include "megiston.h"

...

#include "Rayo.h"

define googology := googol+googolplex+mega+megiston+ ... +Rayo;

The problem? Well if this is a real file it must be static. Think of it like code siting inert in a computer. If there is only a finite number of include files, and your absolutely explicit about what is and is not included ... then fine ... congradulations you've finally invented a well-defined value! (but that doesn't mean you've mastered googology. There is still much to learn, but at least this is a first step) But if the question about what's included is essentially open-ended, and what you really want to say is that your number is ALWAYS dynamically includes anything on the website then that's bullshit. You haven't actually commited to an actual definite honest to goodness number, so you have already violated the 3rd Commandment.

In general, never reference the "whole" of all literature/mathematics/thought space/etc. as part of your number, because none of these are static entities. Don't expect accolades for these kind of cheap tactics. You are not helping to further googology when you do this, you aren't contributing anything new, you are merely incouraging stagnation.

Still not convinced? Alright, there is one final good reason why not to do this. Let's say it's no holds barred. Okay fine. But if you can do it someone else can right? That's only fair.

So you write on a piece of paper "my number is my opponents number +1". Turns out your opponent wrote the same thing. From that day on no body will compete ... it always ends in a draw -_-