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:
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)"
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.
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?
The parent suggested that the number couldn't be encoded due to its largeness rather than its value. So while any number n with Kolmogorov complexity K(n) > 10^100 cannot be recursively encoded in the known universe, that number n need only be 10^100 bits long. On the other hand a number that's too large to be recursively encoded in the known universe would have to exceed BBλ2(10^100),
where BBλ2 is an optimal busy beaver for prefix-free binary programs [1].
Yes, I understood what the parent suggested. I am pointing out that such a number may have strange properties like the fact that a number larger than it can have a smaller Kolmogorov complexity, then I am questioning whether there is a number such that every number larger than it has such a large Kolmogorov complexity that it cannot be encoded. The question therefore becomes, is there a limit to the size of physically describable numbers? Or is there always going to be some number larger with some trivial kolmogorov complexity?
Postulate: You cannot define a largest physically describable number.
My assumption is that due to the very nature of Kolmogorov complexity (and other Godel related / halting problem related / self referential descriptions), this is not an answerable or sensible question.
It falls under the same language-enabled recursion problems as:
- The least number that cannot be described in less than twenty syllables.
- The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than a googol (10^100) symbols.
If you could encode it that way, then it's incoherent. After all, that encoding exists within the universe. If it resolved to a value, that would disqualify that value from being correct because of the self reference.
Not really as that implies that you have a list of numbers that can be encoded within the universe, but the universe would run out of room keeping that list.
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.
Relativistic speeds can contract the length of any object as measured from an outside observer. If an object the size of 1 Planck length travels fast enough: you won't be able to measure it, as from your position it would be smaller than the Planck length as it passes by.
It's not impossible (afaik) for things to be smaller than the Planck length. We just don't have the ability (maybe ever) to measure something smaller than this limit.
Now, good luck finding something the size of 1 Planck length, and also to accelerate it to relativistic speeds.
Frustratingly, attempts to discretize space invariably run into problems with relativity, since they effectively impose a preferred frame of reference. I.e. you can impose a minimum distance, but relativistic length contraction means that observers measure different minima and in different directions.
Apparently, under some of these models, this implies that the speed of light ends up depending on wavelength, lending them to empirical tests. My understanding is that these discrete space models have failed to line up with experiment, at least within the limits of measurement.
That's currently unknown. For all current practical purposes, kind of. The plank length sets a limit on spatial resolution of any information, so a finite region with (universally) bounded entropy per conceivable bucket on that scale still has finite entropic capacity.
The current theories use continuous space and time. However, we can't encode information into space itself. We would have to use some configuration of matter, and then there are limits to how well-defined a particle's position can be coming from the uncertainty principle.
On the other hand, general relativity implies that if you put enough matter in a small enough space it becomes a black hole, and then we can't access the information in it.
IANA physicist but I think this line of thought is probably speculative at the moment because it involves both general relativity and quantum mechanics and it isn't known how they should work together.
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.
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
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".
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.