*Googology Wiki* is an online community and wiki encyclopedia devoted to googology, the study of large numbers and their nomenclature. It boasts 10,974 articles on numbers big and small—some everyday, and some far beyond human comprehension. This wiki was created on December 5, 2008, with the intent of gathering the sporadic large number community into a single place.

## Contents

- 1 Googology
- 1.1 What is googology?
- 1.2 What is Googology Wiki?
- 1.3 Is Googology Wiki peer-reviewed?
- 1.4 Isn't this a waste of time?
- 1.5 Where can I learn more about googology?
- 1.6 Can't you just add 1?
- 1.7 I know the biggest number! Infinity!
- 1.8 What are googol and googolplex? How big are they?
- 1.9 Is googolplex the largest number?
- 1.10 Okay then, is googolplex the largest
*named*number? - 1.11 I heard that googol was coined by a kid. Is this true?
- 1.12 How big is ____?
- 1.13 What's BEAF?
- 1.14 What's the fast-growing hierarchy?
- 1.15 How can I get started creating googology?

- 2 Googology Wiki

## Googology

### What is googology?

**Googology** is an obscure field of mathematics dedicated to studying and naming big numbers. It is a strange cross between a mathematical field, an art form, and an odd hobby. See our article on the subject for a description and a short essay on the history of large numbers.

Large numbers have been part of human society ever since we could imagine them, but the term "googology" and the unification of large number studies are much newer. A Google search for the word returns only about 24,500 results as of September 2014.

Googology has nothing to do with the Google search engine. The study of Google is called "googlology." (Google is, in fact, a misspelling of "googol" that stuck.)

### What is Googology Wiki?

Googology Wiki is an encyclopedic reference, in, discussion forum for large number studies.

### Is Googology Wiki peer-reviewed?

This isn't so much a frequently asked question as something we should clarify straight up. **Googology Wiki is not a peer-reviewed encyclopedia or journal, and it is not subject to professional quality control.** Most editors here are in fact amateurs, and although we try hard to write accurate content, we certainly aren't immune to human error. Like Wikipedia, we try to cite sources (some reliable, some not), so of course you should check those. When sources are not cited, you're dealing with original content. Tread carefully, think critically.

And, since this is a wiki: if you think you've found an error, fix it! At the very least, leave a message at the offending article's talk page to inform other editors of the problem. Any kind of constructive feedback is appreciated.

### Isn't this a waste of time?

Not completely. Large numbers can arise out of much more practical mathematical studies. Some examples are Graham's number (from Ramsey theory), the fast-growing hierarchy (from set theory), and n(4)/TREE(3)/SCG(13) (from combinatorics). Other large numbers were invented purely for entertainment, such as zootzootplex and Clarkkkkson. Overall, googology is recreational mathematics, although sometimes it flirts with more serious topics.

We do googology because it's fun. It's immensely entertaining to toy with concepts that we know we'll never be able to comprehend. Googology is a bit like theology, except you make up your own gods and you can usually prove they exist (unless they're independent of ZFC).

As with many nerdy subcultures, people get mad at us and do the whole "you guys could be curing cancer"/"nobody cares about math" schtick. If you're one of those people, buzz off and go play your World of Warcraft or something.

### Where can I learn more about googology?

We're flattered by your interest! Here are two newbie-friendly sites:

- Sbiis Saibian's in-progress Web book is an excellent introduction to the field, meant to appeal to both novices and hardcore googolsmiths. Sbiis gives you a tour of the history of googology, and gives you an inside view of the googologist's mind as he walks you through the creation of a new large number notation, the Extensible-E system. And yes, this wiki is weakly affiliated with the book.
- Robert Munafo has created an extensive tour of numbers small to big, reaching all the way into Cantor's infinities.

For more sites, see our links page.

### Can't you just add 1?

You can. That's an inherent property of the natural numbers; you can always go further. (Unless you're an ultrafinitist. But ultrafinitists are crazy.) This does *not* mean that googology is a worthless endeavor.

Googology is not just about building big numbers — it's about using the simplest and fastest method possible to build extremely large numbers. Adding 1 is a ridiculously slow method. Why not multiply or exponentiate? Or even better, tetrate?

### I know the biggest number! Infinity!

Googology is the study of large finite numbers. Whereas infinity is certainly large, and it may tentatively be considered a number, it certainly isn't finite! In addition, the word "infinity" can have a variety of meanings in different mathematical fields. Minus context, the concept "infinity" is meaningless, and in no way googological.

One realization of the concept is the theory of **ordinals**, first proposed by the German mathematician Georg Cantor in a system that extends the natural numbers. The smallest infinite ordinal is \(\omega\), which is the least ordinal greater than all the positive integers. Perhaps surprisingly, these infinities are incredibly useful in the study of large finite numbers! See the fast-growing hierarchy.

"Infinity" is defined as "a 'number' greater than any other number." But for any ordinal \(\alpha\) you can always generate a bigger one, so perhaps "infinity" is not a good way to describe the ordinals. Cantor noted this and preferred to use the term **transfinite numbers**. Googologists also use the phrase "absolute infinity" to mean "a mathematical symbol signifying that the race for large numbers will never end." There is no absolute infinity + 1, because absolute infinity isn't a number at all.

### What are googol and googolplex? How big are they?

Googol is 10^{100} = 10 × 10 × ... × 10 × 10 (100 times) = 1 followed by 100 zeroes. For comparison, the number of subatomic particles in the observable universe is estimated to about 10^{80}.

Googol is larger than million, billion, trillion, ... The 33rd member of this series, tretrigintillion = 10^{102}, is the first one to surpass googol. In exact terms, it is 10 duotrigintillion.

It is possible and not too difficult to write out a googol. The author managed to do it in 45 seconds.

Googolplex is 10^{10100} = 10^{googol} = 10 × 10 × ... × 10 × 10 (googol times) = 1 followed by googol zeroes. It is *not* googol + googol or googol × googol or googol^{googol}, the first two being too small and the last one being too big (10^{10102}). I shall say this once more — googolplex is 10 to the power of googol. Nothing else!

Writing out a googolplex in any font size would take up more space than the observable universe can hold.

### Is googolplex the largest number?

Nope! This is a common misconception (and a bizarrely illogical one). Googolplex + 1 is well-defined, and that's enough to show that googolplex is not the largest number. Googologists don't like to "just add 1 and be done with it," though.

### Okay then, is googolplex the largest *named* number?

Nope! There's googolduplex (googolplexplex), giggol, gaggol, grangol, tritri, gongulus, bongulus... and a lot more!

The largest named number as of July 2015 is BIG FOOT, which is so large that its value can't be found even with a computer with finite memory! Before you go out and define BIG FOOT^{100}, do remember that googologists are interested in finding elegant ways to reach new large numbers, not taking existing numbers and adding random junk to them.

### I heard that googol was coined by a kid. Is this true?

Yes. The whole story is a classic legend of mathematical folklore. Googol was coined in the 1920s by a 9-year-old Milton Sirotta when his uncle Edward Kasner, a mathematician and writer, asked for a name for the number. Milton then coined "googolplex"; it was first suggested that it should be 1, followed by writing zeroes until you got tired, but since this is subjective, the googolplex ultimately got defined as 10^{10100}, or 1 followed by googol zeroes. The two number names were perhaps first published in *New Names in Mathematics* (1937).

This is the correct story according to primary sources. Like all legends, it has seen some corruption in retelling.

Googolplexplex came much later, and was probably invented independently by several people.

### How big is ____?

We get a lot of comments asking "how big is Graham's number" or "how big is Rayo's number." Usually, these editors are asking for a concrete description such as \(10^{10^{10^5}}\) and find the encyclopedia's definition to be too difficult. The answer in these cases is that these numbers are so large that they can only be described in abstract terms.

If you don't understand part of the definition, leave a comment on the article's talk page. Don't just say "I don't get it" — please describe exactly what is unclear to you in the article. More knowledgeable users would be happy to help you; just show that you've put some real effort into trying to understand the content.

### What's BEAF?

BEAF stands for Bowers Exploding Array Function. Read the article or the introduction.

### What's the fast-growing hierarchy?

Read the article. If it's too technical, there's an incomplete introduction.

### How can I get started creating googology?

Before you create any new googology, it is absolutely a good idea to familiarize yourself with the existing work. You don't want to find yourself reinventing the wheel, and you should show that you've done your homework when conversing with other googologists.

Sbiis Saibian's Introduction to Googology blog post is an excellent crash course on some of the most common beginner mistakes.

## Googology Wiki

### I made up some names of numbers. Can I post them?

Yes, but **please do so as a blog post**. We're both an encyclopedia and a discussion forum for large number studies, and blogs are a way to get your ideas out to the community (especially if you wish to have things reviewed).

You need to have a registered Wikia account to make a blog. Once you have an account, click on your username in the upper right (or click this link), click on the "Blog" tab, and press "Create Blog Post". (direct link to your blog)

**Do not create mainspace articles on your own work.** We used to allow this as long as the content is cited from an external source, but this has changed.

### How do I create fancy-looking equations?

Googology Wiki uses LaTeX (tutorial), and more specifically MathJax. Use `\(x + 1 = 2\)`

for inline equations and `\[x + 1 = 2\]`

for separate ones. The built-in MediaWiki math system also works, but use it sparingly — specifically, it is useful when MathJax is slow or doesn't load in blog comments.