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.
Ok I think we have a wildly different way of thinking and talking about things. When I talk about "volume" what I mean is what physicists usually discuss - the volume of the universe is the number of 1 cubic meter cubes you can hypothetically fill it with.
In other words what we want to do is divide up the universe into cubic meter boxes, then put those boxes into bijection with some mathematical objects to "count" them. Fairly obviously the number of 1m objects you can fill an "infinite" universe with is larger than any finite number, so we say it is infinite.
This doesn't look like any kind of "potential" process, the number of boxes is literally in bijection to the number of natural numbers, so we say it is infinite.
As an aside, calculus really doesn't deal with infinite quantities at all, at least as its formulated in (e.g.) a modern real anysis course or something. Sometimes we use infinity as a convenient shorthand/slang when describing limiting behavior of functions or whatever, but the actual formal constructions of calculus are entirely about finite numbers. You don't need "potential infinities" at all.
Incidentally the stuff about discrete/continuous space-time very strongly does not matter for this point. I'm dividing your tower up into a countable infinite set of unit volumes whether or not the underlying space-time is discrete or continuous.
You have ignored most of what I have written. I already explained why the potential notion of infinity makes more sense than the actual notion. The reason is that size comparisons of volumes make much more sense with the former.
Now you repeat the definition of actual infinity, which is highly irrelevant, because I already discussed size comparisons, a more advanced topic, which you ignore. Please read my post again.
> This doesn't look like any kind of "potential" process
There is no actual process in time anyway, there is just the property of some quantity being unbounded.
> As an aside, calculus really doesn't deal with infinite quantities at all, at least as its formulated in (e.g.) a modern real anysis course or something. Sometimes we use infinity as a convenient shorthand/slang when describing limiting behavior of functions or whatever, but the actual formal constructions of calculus are entirely about finite numbers. You don't need "potential infinities" at all.
That's completely wrong. You are operating under the assumption that potential infinity is a number. It's not, it's a property of being unbounded. Only actual infinity is a type of infinity that is a number. Or rather a collection of "transfinite numbers" of different size.
> Incidentally the stuff about discrete/continuous space-time very strongly does not matter for this point. I'm dividing your tower up into a countable infinite set of unit volumes whether or not the underlying space-time is discrete or continuous.
If you want to compare sizes under actual infinity, the discrete/continuous distinction matters. Consider the set described by the interval [0, 1]. Is it countable or uncountable? If you are dealing with real numbers, it is uncountable, even if you manage to divide it into only countably many sub-intervals. Arbitrarily dividing things is arbitrary, what matters for size is the underlying structure.
Otherwise the tower would have both countable and uncountable volume at the same time (which is a contradiction), because partitions satisfying either are possible.
I didn't ignore what you wrote, I disagreed with it. It may be clear to you that potential infinity is the way to assess the volume of the towers but I think it is incorrect.
I don't think that volume is the appropriate thing to use for the sort of size comparison you want to make. The argument you made is that we should consider one tower to have twice the volume of the other because we can fit it inside twice. I don't think this is a useful notion, since it is trivial to to fit one tower inside the other tower arbitrarily many times if you slice them up a bit.
I think the sense in which one tower is twice as big as the other is captured by another quantity you mentioned already - the cross-sectional area. You don't need this potential vs actual infinity stuff at all. You just talk about whichever of the volume and cross-section is relevant to you.
> That's completely wrong. You are operating under the assumption that potential infinity is a number.
No type of infinity is a number, but I was indeed working under the assumption that it could be used as a cardinality since you told me you could understand volume with it, and I told you I understand volume as the cardinality of a set of unit-volumes filling the universe. If potential infinity can not be understood as a cardinality it seems entirely impossible to talk about it being the volume of the universe.
> Consider the set described by the interval [0, 1]. Is it countable or uncountable?
Neither, it is finite. We are talking about volumes here (used as short hand for lengths/areas/whatever is appropriate for the dimension), not cardinalities. Its volume is 1. If the underlying space is continuous then its cardinality is going to be uncountable, if the space is discrete the cardinality will be finite (not countable but finite), but either way the volume (defined by the appropriate measure) will be 1.
> Otherwise the tower would have both countable and uncountable volume at the same time
Nope, by fairly standard arguments your tower can be partitioned up into an countable number of unit cubes, and it emphatically can't be partitioned up into an uncountable number of unit cubes. You can use essentially the same argument that says one can't partition the real line up into uncountably many unit intervals.
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.