Comment by xelxebar
6 months ago
Pentation? How quaint. For other Large Number fans, David Metzler has a wonderful playlist that goes way down the rabbit hole of the fast growing hierarchy:
https://www.youtube.com/playlist?list=PL3A50BB9C34AB36B3
Highly recommended.
Almost all of these mind-bogglingly large numbers are built around recursion, like the Ackermann function, which effectively has an argument for the number of Knuth up arrows to use. Then you can start thinking about feeding the Ackermann function into that slot and enjoy the sense of vertigo at how insane that becomes.
I find it fascinating how quickly the machinery of specifying large numbers probes the limits of what's definable within the mathematical systems we use.
It's funny how, for a certain subset of math, a researcher's life can be condensed to "arguing about which number is the biggest (preschool)" -> "learning about math" -> "arguing about which number is the biggest (academia)"
Numberphile has also built-up a fantastic collection of videos over the years about unfathomably large finite numbers:
https://youtube.com/playlist?list=PLt5AfwLFPxWJ_FwchNAjB3xa8...
The tools required to probe and compare these monsters are so interesting.
This too:
https://www.mrob.com/pub/math/largenum.html
Cool wiki linked there: https://googology.fandom.com/wiki/Googology_Wiki
Reminds me of the Scott Aaronson essay ["Who Can Name the Bigger Number"](https://www.scottaaronson.com/writings/bignumbers.html)
There is something awesome about incredibly large finite numbers. People gush about infinity, but I find it to be a pretty boring concept compared to finite numbers too large to even be written in this universe.
Infinity is aspirational. Infinity is a concept, simple and self-evident, yet paradoxical and oxymoronic.
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.
I'm wondering if there's a connection between large number hunters, unwritten rule proponents in sports and games, and modular synth collectors. There's a sort of satisfaction derived from finding and defining order according to aesthetic structures that are largely arbitrary but also emotionally resonant.
Meanwhile, infinity is for those that embrace chaos, minimalism, nothingness.
> People gush about infinity, but I find it to be a pretty boring
You need to look at this:
https://neugierde.github.io/cantors-attic/
Yeah!
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 think such a number is going to have strange properties like, some number bigger than that unencodable number is encodable because of a special pattern that allows a special non-surjective recursive function to encode it. I am just wondering if there really is smallest number for which no number greater than it is encodable.
It is not obvious to me that the definition of an encodable function has bounded growth: is it true that f(1) - f(0) for encodable f always has a bound given by the amount of data used to encode f? What is that bound?
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but couldn't you encode it as "the smallest number that cannot be encoded within the universe's matter/entropy"?
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There is enough information if you assume reality is continuous. Pick a point A to be the origin. Then then you can encode the number by placing something at 1/N meters away from the origin.
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For me it is far more to consider the number of atoms in the solar system, as it implies a pretty obvious limit on the data storage that humanity can meaningfully use. Obviously only a tiny fraction of those atoms can be used to store information & furthermore the energy required to actually perform that storage is enormous compared to the energy we have available at present.
Would you like to also be able to communicate about this number? You might have to reserve some atoms to form a being that could actually enjoy it. Considering such a perspective, the decoder for observing it should probably be embedded in the number itself.
You are stretching the definition of "is" here. Almost all finite numbers are impossible to actually count up to, so they exist only in the same sense that infinite numbers exist.
Indeed! Funnily enough, there seems to be something similar going on between defining large finite numbers and defining large countable cardinals.
And I’m assuming they can be converted to incredibly small finite numbers just as easily?
You might already know this, but the busy beaver function grows faster than any computable function. So although the best known lower bound of BB(6) can be expressed with just pentation, generally speaking the BB function is certainly beyond any standard operation in terms of fast growth
Tree(n) is considered computable.
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).
It's been shown to be surpassed at n=150, which as you note is likely very generous. Hypertree would typically only require a few more states. Hypertree doesn't grow meaningfully faster than Tree. Hypertree(3) is what would be called a Salad number, combining a very fast growing function with some very weak one(s) such as iteration, which in the Fast Growing Hierarchy corresponds to taking the successor if an ordinal.
The growth of BB is certainly mind-boggling; however, I personally find its growth rate so untouchable as obviate any attempt at understanding. There's nothing to gain purchase on.
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.
If you have a child who likes math I highly recommend "Really Big Numbers" by Richard Schwarz. Tons of nice illustrations on how to "take bigger and bigger steps".
"Infinity is farther away than you thought."