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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 num

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.

xelxebar 1y ago
Indeed! Funnily enough, there seems to be something similar going on between defining large finite numbers and defining large countable cardinals.
Waterluvian 1y ago
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?
- charcircuit 1y ago
  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.
  - para_parolu 1y ago
    Layman question. But can space be quantinized (not sure what is proper term)? Like there is finite positions for particle between two points?
    - hansvm 1y ago
      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.
    - xelxebar 1y ago
      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.
    - griffzhowl 1y ago
      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.
  - moi2388 1y ago
    No, because you can’t have something have that precisely defined of a position.
    - chii 1y ago
      you can - you merely need enough energy to precisely define it (as according to the heisenberg uncertainty principle)! Unless...such an energy exceeds the Schwarzschild radius...
      - moi2388 1y ago
        So you can’t.
      - close04 1y ago
        You can't halve a Planck length so you're limited to ~1.6×10^−35.
        lvncelot 1y ago
        I think current theories break down at less than a Planck length, but they are not constrained to integer multiples of it.
        altruios 1y ago
        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.
    - charcircuit 1y ago
      By definition you can if you accept it's continuous.
  - deleted 1y ago
    [deleted]
- chii 1y ago
  but couldn't you encode it as "the smallest number that cannot be encoded within the universe's matter/entropy"?
  - ndsipa_pomu 1y ago
    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.
  - Scarblac 1y ago
    What's the decoding algorithm for that?
  - recursive 1y ago
    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.
- BlackFly 1y ago
  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?
  - tromp 1y ago
    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]. [1] https://oeis.org/A361211 https://oeis.org/A361211
    - BlackFly 1y ago
      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?
      - ccvannorman 1y ago
        I absolutely love this question. 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.
- weregiraffe 1y ago
  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.
- sidewndr46 1y ago
  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.
- smokel 1y ago
  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.
dr_dshiv 1y ago
And I’m assuming they can be converted to incredibly small finite numbers just as easily?
lisper 1y ago
> 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/ https://neugierde.github.io/cantors-attic/
jrowen 1y ago
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.
- jrowen 1y ago
  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.

xelxebar 1y ago

Indeed! Funnily enough, there seems to be something similar going on between defining large finite numbers and defining large countable cardinals.

Waterluvian 1y ago

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?

charcircuit 1y ago
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.
- para_parolu 1y ago
  Layman question. But can space be quantinized (not sure what is proper term)? Like there is finite positions for particle between two points?
  - hansvm 1y ago
    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.
  - xelxebar 1y ago
    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.
  - griffzhowl 1y ago
    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.
- moi2388 1y ago
  No, because you can’t have something have that precisely defined of a position.
  - chii 1y ago
    you can - you merely need enough energy to precisely define it (as according to the heisenberg uncertainty principle)! Unless...such an energy exceeds the Schwarzschild radius...
    - moi2388 1y ago
      So you can’t.
    - close04 1y ago
      You can't halve a Planck length so you're limited to ~1.6×10^−35.
      - lvncelot 1y ago
        I think current theories break down at less than a Planck length, but they are not constrained to integer multiples of it.
      - altruios 1y ago
        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.
  - charcircuit 1y ago
    By definition you can if you accept it's continuous.
- deleted 1y ago
  [deleted]
chii 1y ago
but couldn't you encode it as "the smallest number that cannot be encoded within the universe's matter/entropy"?
- ndsipa_pomu 1y ago
  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.
- Scarblac 1y ago
  What's the decoding algorithm for that?
- recursive 1y ago
  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.
BlackFly 1y ago
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?
- tromp 1y ago
  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]. [1] https://oeis.org/A361211 https://oeis.org/A361211
  - BlackFly 1y ago
    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?
    - ccvannorman 1y ago
      I absolutely love this question. 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.
weregiraffe 1y ago
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.
sidewndr46 1y ago
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.
smokel 1y ago
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.

dr_dshiv 1y ago

And I’m assuming they can be converted to incredibly small finite numbers just as easily?

lisper 1y ago

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

jrowen 1y ago

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.

jrowen 1y ago
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.