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Could you explain how a potentially infinite universe would be different to an actually infinite one? I'm not familiar with how you're using these terms.

nairboon 3y ago
I don't know what the difference would be. That's why I'm asking. These terms regained popularity due to Cantor's work. The potential infinite, is a very old idea for a process that never terminates. Like counting, you can always find a bigger natural number, by addition. Thus, the natural numbers are potentially infinite, because they grow forever, but it will never be finished. However, Cantor popularized another notion of infinity, namely that you can treat the ever growing natural numbers as one finished big bag and started doing math with this "object". Cantor's theory requires that infinity actually exists, as a finished object. Then he starts doing interesting stuff, as in measuring how many things are in the bag, even though the bag technically grows without ever stopping.
- eigenket 3y ago
  Ok, well if you come up with a definition for what a potentially infinite universe means then let me know. Until then our best models have the universe being infinite in size in the normal sense of the word.
  - nairboon 3y ago
    > being infinite in size in the normal sense of the word. Well, can you maybe define what this normal sense of this word is? This is a thread about Cantor, whose whole work is based on subtle differences in the treatment of "infinity".
    - eigenket 3y ago
      I apologise, since you're the one who brought up the idea that the universe was either potentially or actually infinite I thought you might have some idea what those terms meant. By "normal" or "actual" here I meant how a physicist would use the term - an infinite universe is one which can contain objects of any finite volume (if you like, can contain onjects of any finite volume larger than some minimum). I presumed (perhaps incorrectly) this is how you were using the term acutal, but now I see I should not make such assuptions.
      - cubefox 3y ago
        Potential infinity is the notion of infinity that is e.g. used in calculus. Potential infinities are compared by rates of growth relative to some common quantity, while actual infinities are compared by the possibility of defining a surjective mapping of objects in one to objects in the other (which is treated as ≤). So when we talk about a universe that may be "infinite" in size, do we talk about a "potential" or an "actual" infinity? I think it's clearly the former. If we talk about the size of the universe, we seem to talk about a geometric volume. Now think of two "infinitely" tall towers standing on the ground. Tower A has a cross section of 100m², while tower B has 200m². Assume that the cross section of tower A has rectangular shape and of tower B square shape, such that you could fit exactly two of tower A into tower B if the latter was hollow. This suggests a clear meaning in which tower B has "twice as much" volume as tower A, even though both have "infinite" volume. You can literally fill tower B exactly with two towers A. Here "twice as much" just means that as you go higher, the volume of tower B increases twice as fast as for tower A. Which is the rate-of-growth size-comparison from potential infinity. But for actual infinity you can't say tower B has twice as much volume. You are forced to assume their volume is the same. But you don't even know whether their volume is "countably" or "uncountably" infinite, since you don't know whether space-time is quantized (countable) or continuous (uncountable). But that doesn't even matter for comparing the volume of the two towers: It's not sensible to say the towers would gain volume by switching from a discrete universe to a continuous universe. Discrete vs continuous is only about how far space can be divided, which is independent of its volume. Otherwise we also would have to say that 1m³ in a continuous universe is more volume than 1000m³ in a discrete universe. Which would be plain wrong, 1m³ is less volume than 1000m³, no matter what the microstructure of space is like. And if we say the universe is infinitely large, we apparently just talk about its volume (or hypervolume of space-time etc). Which would e.g. mean that an infinite universe with less dimensions would literally fit inside a universe with more dimensions, but not the other way round. (Assuming the physical laws are otherwise the same.) So, it seems clear that an "infinite universe" assumes the notion of potential infinity, not of actual infinity. I don't know anything about the supposed infinities present in a hydrogen atom, but I would guess that those probably are potential infinities, too. Which would suggest that the notion of an actual infinity is not backed by physics of the real world.
ludston 3y ago
A potentially infinite universe is also potentially finite? I.E. There is not sufficient evidence to determine that it is infinite since we are only able to make observations of a finite distance?

nairboon 3y ago

I don't know what the difference would be. That's why I'm asking. These terms regained popularity due to Cantor's work. The potential infinite, is a very old idea for a process that never terminates. Like counting, you can always find a bigger natural number, by addition. Thus, the natural numbers are potentially infinite, because they grow forever, but it will never be finished. However, Cantor popularized another notion of infinity, namely that you can treat the ever growing natural numbers as one finished big bag and started doing math with this "object". Cantor's theory requires that infinity actually exists, as a finished object. Then he starts doing interesting stuff, as in measuring how many things are in the bag, even though the bag technically grows without ever stopping.

eigenket 3y ago
Ok, well if you come up with a definition for what a potentially infinite universe means then let me know. Until then our best models have the universe being infinite in size in the normal sense of the word.
- nairboon 3y ago
  > being infinite in size in the normal sense of the word. Well, can you maybe define what this normal sense of this word is? This is a thread about Cantor, whose whole work is based on subtle differences in the treatment of "infinity".
  - eigenket 3y ago
    I apologise, since you're the one who brought up the idea that the universe was either potentially or actually infinite I thought you might have some idea what those terms meant. By "normal" or "actual" here I meant how a physicist would use the term - an infinite universe is one which can contain objects of any finite volume (if you like, can contain onjects of any finite volume larger than some minimum). I presumed (perhaps incorrectly) this is how you were using the term acutal, but now I see I should not make such assuptions.
    - cubefox 3y ago
      Potential infinity is the notion of infinity that is e.g. used in calculus. Potential infinities are compared by rates of growth relative to some common quantity, while actual infinities are compared by the possibility of defining a surjective mapping of objects in one to objects in the other (which is treated as ≤). So when we talk about a universe that may be "infinite" in size, do we talk about a "potential" or an "actual" infinity? I think it's clearly the former. If we talk about the size of the universe, we seem to talk about a geometric volume. Now think of two "infinitely" tall towers standing on the ground. Tower A has a cross section of 100m², while tower B has 200m². Assume that the cross section of tower A has rectangular shape and of tower B square shape, such that you could fit exactly two of tower A into tower B if the latter was hollow. This suggests a clear meaning in which tower B has "twice as much" volume as tower A, even though both have "infinite" volume. You can literally fill tower B exactly with two towers A. Here "twice as much" just means that as you go higher, the volume of tower B increases twice as fast as for tower A. Which is the rate-of-growth size-comparison from potential infinity. But for actual infinity you can't say tower B has twice as much volume. You are forced to assume their volume is the same. But you don't even know whether their volume is "countably" or "uncountably" infinite, since you don't know whether space-time is quantized (countable) or continuous (uncountable). But that doesn't even matter for comparing the volume of the two towers: It's not sensible to say the towers would gain volume by switching from a discrete universe to a continuous universe. Discrete vs continuous is only about how far space can be divided, which is independent of its volume. Otherwise we also would have to say that 1m³ in a continuous universe is more volume than 1000m³ in a discrete universe. Which would be plain wrong, 1m³ is less volume than 1000m³, no matter what the microstructure of space is like. And if we say the universe is infinitely large, we apparently just talk about its volume (or hypervolume of space-time etc). Which would e.g. mean that an infinite universe with less dimensions would literally fit inside a universe with more dimensions, but not the other way round. (Assuming the physical laws are otherwise the same.) So, it seems clear that an "infinite universe" assumes the notion of potential infinity, not of actual infinity. I don't know anything about the supposed infinities present in a hydrogen atom, but I would guess that those probably are potential infinities, too. Which would suggest that the notion of an actual infinity is not backed by physics of the real world.

ludston 3y ago

A potentially infinite universe is also potentially finite? I.E. There is not sufficient evidence to determine that it is infinite since we are only able to make observations of a finite distance?