3.3.1. The Curious Paradox of Fast-Growing Sequences

An Overview of Sequences

If you've ever been really interested in learning about numbers, chances are you know your way around sequences. Sequences of numbers come in all shapes and sizes. Everyone knows about prime numbers, which have a simple definition but behavior that just barely eludes any form of logic or consistency. Everyone also knows of the Fibonacci numbers; those start with a cute gimmick but gradually become a mind blow as the ratio between each number asymptotically approaches the legendary golden ratio, a staple of nature and architecture alike that has been recognized since ancient times. Or how about the perfect numbers, a peculiar set of numbers that when unraveled reveals itself to be tied to the Mersenne primes, which are themselves tied to the powers of two? And that's just scratching the surface. Sequences are an absolute staple of recreational mathematics. Go to part 1 of my number list, CTRL-F "sequence", and prepare to be lost in a world of wonders. But please note that most of those I stole from Robert Munafo's list. Also, if you have any homework to do please for the love of god do not look up integer sequences.

Most of those sequences of numbers are very silly and may lead to astronomical numbers like a quintillion if you're lucky. But some sequences lead to INSANELY huge numbers! Many of those numbers dwarf Graham's number; some surpass the breaking point of epsilon-zero in the fast-growing hierarchy. These sequences usually arise from graph theory or simple "games" that turn out to take a monstrous time to solve, and are a uniquely fascinating part of googology that I have a bit of an odd relationship with. Crazy numbers that arise from those sequences are actually connected to other parts of math; why don't people flock towards those instead of silly large number notations? To answer that, let's talk about Graham's number.

Graham: A Tale of Theory vs. Practice

The size of Graham's number is fascinating and all, but sometimes I find myself much more intrigued by the story behind it. I explained the math problem Graham's number originated from in this article; I tried to made my explanation as accessible as possible. I find it a shame that so many people talking about Graham's number just say it originated from "some math problem", assuming by default that it is boring and insanely complicated. Do those people not understand any bit of the beauty of mathematics? That simple elementary concepts can lead to absolute insanity??? Or do they just have PTSD from being forced to integrate wild jumbles of trigonometric functions? Maybe those two are synonymous.

Conventional wisdom holds that Graham's number is the largest number that has found a use outside of just "making large numbers for fun", which as I established prior is false on so many levels. Ronald Graham's 1971 paper about Ramsey theory was home to a number Sbiis Saibian calls "Little Graham", an absurdly high upper bound for a Ramsey theory problem. Famed math writer Martin Gardner brought Graham's paper to the public eye; he saw potential in that number as a subject that will wow readers, so he and Graham worked to devise a more accessible number that can be just as well proven to be an upper bound to that problem. And it worked spectacularly! Introduce up-arrow notation, create 3^^^^3 and dub it G1, iterate the number of up-arrows onto itself until you get G64; that number is now known as Graham's number and it's really a bit of a fabrication. It was created as a convincing number that is provably greater than a number that is itself provably greater than some mystery value g (minimum number of dimensions for a hypercube to always meet Graham's criteria). And we now know that g is in fact lower than 2^^2^^2^^9, a modest pentational number. It could be as small as 13!

So why does Graham's number get more attention than TREE(3) and all those other numbers that are not upper bounds, but real solutions to problems? This question perplexed me years ago, but now the answer is obvious. Graham's number holds a special charm in being both easy to explain and easy to blow your mind, as well as the label attached that it was "used in a real math problem". Many enormous numbers crop up when looking at hard-to-solve mathematical problems, but you have to be lucky to devise one whose size hits that sweet spot. As such, large numbers arising from math problems instead of being deliberately crafted with systematic notations are a dark spot in the googology community. To get the attention they deserve, they need to fall into some sweet spot.

TREE(3): Graham's but Better(?)

TREE(3), a number devised by Harvey Friedman (yes, he's the guy I met at an Ohio State University event when I was 10), is the ringleader of this mysterious corner of googology: large numbers that are output by sequences. Some of the sequences that output such large numbers arise from graph theory; some from simple "games" that turn out to take a monstrous but finite amount of time to solve. Many consider TREE(3) the number that should be getting all the attention Graham's number gets. I wholeheartedly agree in theory; both arise from problems that aren't too difficult to understand. But there's one caveat: TREE(3) is a bit too big.

What do I mean by too big exactly? Well, as I said before Graham's number falls into a sweet spot where its size blows people's minds and its computation is easy to explain. TREE(3) falls way beyond that spot. Nobody can give a size that does this number justice unless they use some insanely complicated notation. Same goes for most other numbers in that whole era. Even Little Graham isn't the easiest to explain; Gardner was lucky that he could save that number by making a bigger but easier to explain one.

I find it upsetting how little attention these numbers that come from sequences get. I think more emphasis should be placed on the stories behind those numbers. The story behind Graham's number isn't that complicated, and neither is TREE(3)'s. If I were to write articles on these numbers that come from sequences, I'd focus much more on how they came to be than their size. It doesn't help that I have absolutely no idea how people get to these insane values. I deeply respect anyone with the patience to sift through technical stuff and pages upon pages of formulas so that they can properly understand these crazy results. I'm just not that kind of person. This site was always meant to be beginner-friendly.


I consider fast-growing sequences to be a paradox in googology. They should logically be the part that the community gives the most attention to, but a lot of those are slept on because the large outputs aren't specifically crafted to be impressive numbers. I hope that this aspect of googology will one day get more attention, or have more of its aspects explained in beginner-friendly ways.

... So yeah. This post sure exists. It's the first substantial content on this site in over three years. I hope that I've made my views and potential future plans clear through this page. I get lots of enjoyment out of explaining beginner or intermediate googological concepts in exciting ways, or going on and on about the purely whimsical. But running through super complex math like this while staying enjoyable to read requires a special skill that I'm afraid I don't have.

Don't rule out the possibility of me writing articles about particular sequences that generate large numbers! But most of my writing I currently publish on my Blogger. It's mostly dissecting works of media and rambling about my grievances with myself though; maybe not a googologist's cup of tea. It also may or may not have an absurd amount of posts about a certain webcomic.