Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math

Like, there is a perfectly finite number, but is so large that there simply isn’t enough information in the universe to encode it in any format. How cool is that to just think about for a while?

I get kind of intrigued by these large number things at first but ultimately it feels like kind of a less elegant version of the same thing? It's all this mucking about with multiples and powers of multiples and powers when it's like...we've already taken this to the limit, we can just stand back and marvel at that, what are we looking for? We already know you can always add another digit, why invent more and more complicated ways to do that?

This isn't meant to be contradictory to what you're saying or anything, just interesting to explore these different perspectives on what mathematical concepts capture the imagination.

You need to look at this:

https://neugierde.github.io/cantors-attic/

https://www.mrob.com/pub/math/largenum.html

BB(n) surpasses Tree(n) by - at most - when n=2645.

And likely shortly after BB(100).

Now consider the following definition for an exponentially faster growing number:

HyperTree(n) is where the Tree function is nested x times, where x is the result of tree(n).

HyperTree(3) = Tree(Tree(Tree..{Tree(3) long}...Tree(3))))...)

BB(X) would (should) still outpace HyperTree(x) at some value of x.

I don't know where to begin even to give an upper or lower bound of when BB(x) is larger than HyperTree(x).

The fast growing hierarchy, on the other hand, provides oodles of structure for us to explore, and we can construct numbers that are vastly larger than anything BB(6) is likely to hit. In fact, this is why we use the fast growing hierarchy to approximate really big numbers all the time!

When we take something like f_Γ_0 and try to unpack even just the barest surface of its size, I get a feeling of vastness similar to those videos of diving endlessly into fractals.

Anyway, this article seems fine to me. The "exponentiation" comment seems like a bizarre misreading. The article is just trying to explain how big BB(6) is. Before that, it explains what BB(n) is. To think it's solely about exponentiation you have to skip that entire section.

The problem with this particular article is simple: busy beavers numbers aren't interesting because they're big. They don't break mathematics because of that; you can always say "+1" to get a larger number. There's also nothing particularly notable about Knuth's up-arrow notation, which is essentially a novelty that you're never gonna use. Instead, the numbers are interesting because they have fairly mind-blowing interactions with the theory of computability.

All this talk about exponentiation, tetration and pentation have their roots in Peano arithmetic, which for reasons of logical clarity, defines precisely one function -- the "successor function":

s(n) = n+1

Using just this function, Peano defines addition, multiplication and exponentiation with great clarity. Since Peano's time, people have been making adjustments, like including zero among the counting numbers, and by extending exponentiation into tetration, pentation and a few more exotic operations fully discussed in the linked article and elsewhere.

I personally would have liked to see a more complete exposition of the logical roots of the Busy Beaver challenge, and I think the missing parts would have made the article a better read, even for non-specialists. Maybe especially for those readers.

But Quanta articles are perfectly suited to their audience, people who may be inspired to look for more depth elsewhere.

you gotta pay for that. or dig through the academic publisher archives; which you also gotta pay for unless you believe in digital piracy and evil copyright infringement which may or may not fund terrorism

like they used to say: information wants to be expensive so pay to be free

What's the point of your comment? To suck the joy out of everything?

Turing machines are a fundamental object of study within the theory of computation. The complexity and wild behavior that arises from even the simplest machines is a cool discovery. BB(6) was thought to be a lot smaller, but it turns out to be really huge. The Busy Beaver game is also interesting to those who work on combinatorics and theorem provers. And of course many many people in the space of recreational math & volunteer computing love this challenge.

You don't like it? Ok, then. You don't have to.

I consider TREE to be medium sized in the googology universe. It obliterates Graham's Number of course, as well as Goodstein sequence and simple hydras. But it's easily bested by an ingenious matrix generalization of hydras known as the Bashicu Matrix System, even when fixed at 2 rows. 3-row BMS goes far beyond that, let alone n-row. BMS is so simple that it can be encoded in under 50 bytes [1], where TREE would take significantly more. Beyond that there vastly greater numbers still such as Loader's Number. I shouldn't mention Rayo's because that's no longer computable in the usual sense.

[1] https://codegolf.stackexchange.com/questions/18028/largest-n...

There is no case where no method whatsoever will work. It's true that for any method, there are cases where it fails but it's not true that there exist cases for which every method fails.

That's obviously incorrect; the program either halts or it does not. So if you had function that always returned true and another that always returned false, one of those functions would correctly tell you whether a given program will halt — for every program that could possibly exist.

The hard part is figuring out which function to use :)