Googology is the study of large numbers. It basically is to maths what Supertone and Hypertone are to music-theory. Here’s how to get into it.
Recommended study order
This section is intended as a roadmap to the most important and interesting Googological ideas. There is a lot of Googology out there, much of it leads to dead ends. The following is a somewhat direct route to “understanding” (as best as one can) the largest numbers. Knowledge of the previous steps will help in understanding the next steps, and where it does not: it will heighten ones appreciation of the later ideas.
For example, TREE, though widely covered and pretty well-known, is far down the list as it is difficult to appreciate the size of such a vast number without an understanding of the FGH and somewhat-advanced transfinite ordinals for comparisons. Rayo’s number has also been covered by mainstream Youtube channels such as Numberphile, and again the “comparisons” (using the term loosely) given did not do the number itself justice, and instead focused on the input (10^100, trivial) rather than the output (the important number itself). This is not necessarily a case of Numberphile being lazy, rather it is an issue of not having enough time to build up a scale for comparison.
Take your time learning the steps. Meditate on them, drag the process out. There is no rush, just let your understanding build and build at its own pace.
You should have covered these in school, make sure they’re fresh in your head.
- Counting, addition and multiplication
- Squaring, cubing
- Standard form (aka ‘Scientific notation’)
- Factorials (especially 52!)
Fun number names that can be summoned in casual conversation: ‘ugh, I’ve already told you a Centillion times…’
- ‘Million’, ‘Billion’, ‘Trillion’, etc.
- Long-scale vs short-scale billions.
- ‘Milliard’, ‘Billiard’, etc.
- SI Prefixes (e.g. Mega, Giga, and more!)
- Myriagon and Megagon
- ‘Centillion’, ‘Millinillion’.
- The Googol
- Googolplexplex AKA Googolduplex
- Googol-n-plexes (e.g. Googol-100plex, Googol-googolplex). Just make up your own, tbh!
Learn new ways of generating MUCH bigger numbers systematically.
- Hyper-E notation
- Pentation, Hexation, etc.
- Knuth’s up-arrow notation
- Fast Growing Hierarchy (FGH) basics (finite levels)
- Diagonalising the FGH (level ω).
- FGH level ω+1 and ω+2, ‘Graham’s Number (G_64)’ example
Rather than finding the largest (finite) number, this section is about finding the ’last’ number, i.e. the number after all other numbers. And also, here (unlike in normal googology) we can talk about infinities. We can use these infinite numbers to create normal finite numbers using our Fast Growing Hierarchy. As you learn about these numbers and what they mean, it is always a good idea to experiment with combining previous ideas, and seeing what happens when we use these ordinals to diagonalise. Really take the time to reflect on what these ordinals mean in the context of the FGH. You’ll find that things get incomprehensibly fast-growing long before even f_ε_0.
- ω: ’little omega’
- ω+1, ω+2, …, ω+ω
- ω2, ω3, …, ω^2
- ω^n, ω^ω, ω^ω^ω, …
- ε_0, apply addition, multiplication, etc. as we did for ω.
- ε_1, ε_2, …, the ’epsilon numbers’.
- ε_ω, ε_ω+1, ε_ε_0, etc.
- ζ_0: Cantor’s ordinal (and the other ‘zeta numbers’).
- Veblen hierarchy and Γ_0: the Feferman–Schütte ordinal.
- Three-argument Veblen function, other Γ-numbers.
- Extended Velben functions, Small Veblen Ordinal (SVO).
After this point it can be pretty difficult to play with diagonalising the ordinals. Indeed, there’s already multiple ways to interpret f_Γ_0(n). However, the important ideas are consistent across interpretations regardless of the specifics of how we imagine they should be diagonalised.
- Ordinal Collapsing Functions (see how these can be used to express the previous ordinals).
- Large Veblen Ordinal, LVO (much easier to understand with OCFs than with the usual definition).
- Bachmann-Howard Ordinal and beyond
Transfinite ordinals continue beyond the BHO, but you don’t really need to know about them beyond this point to have a grasp of finite Googology. It isn’t clear, for example, how to diagonalise something non-recursive like the Church-Kleene Ordinal. There have, however, been efforts to extend the FGH to include such ordinals, but that’s really big-brained stuff beyond the scope of this article.
Trees and Hydras
Back to the finite: investigate TREE and the rest of the TREE sequence of numbers: TREE = 1, TREE = 3, and then the sequence grows unbelievably rapidly. According to the Googology wiki, TREE > f_SVO(G_64), if you’ve studied the above topics, then you’ll appreciate just how spectacular the TREE sequence is. It is unclear how high-up the FGH we’d have to go to reach a comparable growth-rate for the TREE sequence as a whole.
Even faster-growing sequences can be found with ‘Hydra-games’ and other stuff from Graph Theory.
Limitations of languages
Maths usually is about using logic to build-up rules (like addition and multiplication) that match and explain processes in the physical world. If we’re just looking to produce large numbers, then it does not make much sense for us to limit ourselves to rules inspired by physical reality, or even non-realistic rules (like hexation) that are built upon these realistic rules (hexation is ultimately just a huge amount of exponentiation). To achieve these numbers, we strip maths back down to its building blocks and see how powerful it can be without any lame real-world rules. Needless to say, they produce far bigger numbers than any of the finite numbers discussed so far.
So forget about ‘addition’ and ‘multiplication’. Literally, just forget basic numeracy skills. Becoming conventionally-innumerate is the best way to truly relate to these numbers, they exist in a vacuum-world where most mathematical truths are replaced with whatever’s most convenient.
Basically these boil down the largest finite numbers definable in a given mathematical language with a limited amount of space/rules available.
- Busy Beaver games, confirmed BB(n) numbers and current champions.
- Oracle machines
- Rayo’s number
- Large Number Garden Number
Things get very difficult to follow from Rayo’s number onward as the limits of languages typically need to be formalised in another more fancy and abstract language. LNGN, from what I could understand of the related literature, was defined by systematically defining infinitely many increasingly advanced formal-languages and then diagonalising across ALL of them (like in the FGH). Basically, the final language used is not just fancy and abstract: it is built on infinite layers of fanciness and abstraction. Admittedly, I do not truly understand the number yet, so read into it for yourself.
As my brother put it, very humorously: ‘Imagine we have a really good language and we make a really big number with it’. That is basically what Googology means at the highest-levels.
I have tried writing my own self-contained guides a few times. Unfortunately, there is SO much to cover that it becomes too much work to maintain a high-standard of writing and originality (if I’m just rephrasing what other people have already said there isn’t much point). So instead, I’m going to recommend some websites. Please note that I have not read everything that these sites have to offer.
- Large Numbers by Robert Munafo: A lot of googology covered here in a lot of detail.
- Googology Wiki: Yes
- Tetration.org: Goes into much more detail on tetration and other iterative functions, extending the ideas to complex numbers and more.
- Χάος: Various writings on tetration, googology and the philosophy of mathematics.
- Hedrondude (archive): The famous Jonathan Bowers. He has numbers with his own notations and naming systems if you want something different.
- Large Number Site by Sbiis Saibian: huge website, more DIY-notation and fun “googologisms” to be found.
- P進大好きbot blog: Architect behind the largest large number: The Large Number Garden Number.
- Extremely Large Numbers Playlist: I find the later FGH examples and Ordinal Collapsing Function examples very fun to follow. The comments suggest that there may be some minor mistakes to be aware of.
- List of common mistakes in Googology: Amateurs BTFO’d by Japanese precision-chad. Corrects many intuitive but incorrect assumptions.
- Cantor’s Attic: Small wiki focused on the infinities. Some infinities are bigger than others.
I have listed some more amateur sites and wikis, some of which may contain some misinformation. Do check what you read and, if possible, avoid relying on casual intuitive explanations. Learn precise mathematical definitions wherever possible before building on further ideas. Very often this is worthwhile, and the numbers and functions (once fully understood) are far more impressive than how you would otherwise have thought them to be. Generally speaking, it is wise to steer clear of Youtube videos and mass-appeal news-site articles.