In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.
Our best physical models are just that - models. All models are wrong, some are useful. Infinity in physics is usually a shorthand for "it's so big I don't have to care about the edges"
I guess as religions go, that one is no worse than many, but there isn't really any evidence for it now. Space looks really infinite, the hydrogen atom spectra are very well described by infinite series.
Can you actually show us one of these apparent actual infinities? All one can reasonably postulate is a potential infinity, but this is completely different from Cantors actual infinities.
As far as I know, we can see/measure that the universe is about 93 billion light years across. We literally can't know what's outside, so we don't know how much bigger it is or if it's infinite.
A decent mental model is to think about an infinitely long ruler or measuring tape, where I have positions marked every (e.g.) 1 meter interval. You can imagine stretching or shrinking the ruler which would move the marks further or closer to each other respectively.
If you think about a "cosmology" for this ruler where I keep stretching it further and further forever then the points I have marked will ger further and further away, but the ruler itself is always infinite.
How's that not also a religion? :p
The best we can say is that if the universe is not infinite it appears to be doing a very good job of pretending to be infinite.
(1) for a good idea of what "weird" means here imagine a 2d 1km by 1km square with geometry like Pac man lives in (toroidal), so if you go off the top of the square you appear at the bottom, and if you head off east you appear on the west. Now start at the middle and leave a rock at your current position. Head due east until you cycle round and hit your rock again. You'll have walked 1km (500m to the east edge and 500m from the west edge back to the middle). Now do the same experiment but walk north-east. You'll hit your rock after sqrt(2) km of walking (1/sqrt(2) takes you to the north east corner and the same to get back to your rock).
In other worlds the pac-man torus space is not isotropic some angles are special and more important than others. Essentially the same thing happens in other finite flat geometries you can invent.
https://arxiv.org/abs/1605.07178
This study is (as far as I'm aware) the state of the art on this topic, and based on the CMB observations they use it appears the universe is incredibly close to isotropic. Note that because it is based on CMB data this sort of study is sensitive to what shape the universe was a long time ago when, if it is finite, it was a lot smaller.
Was it shown that the volume in this models are actually infinite or just potentially infinite?
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.
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".
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.
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.
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.
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 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.
If the Old Greeks [1] haven't been able to solve this problem I'm not sure we'll be able to do better than them.
Or is it more helpful if I say I have 10^27 atoms in my body? Quarks? Gluons? Strings from string theory? Still finite number, not "biteable", not infinite.
That’s useful, but very vague, and very movable criteria for “exists”. (Approx. Must be based in physical reality + enough people subjectively agree on it)
Do following finite numbers exist: -1, 0, 0.5, PI, 2^300 (more than particles in observable universe), sqrt(-1)? Do individual digits exist? If PI exist, how many digits does it have? Do models and algorithms in general exist? Do model existance depend on limits of your/somebodies capacity to understand them?
I would propose alternative, but useful way to look at this. “Two” and “infinity” are models. Both these models are useful, but “two” is just more common one. (Still, various infinities are useful for bunch of people)
More to the point, if we're not able to decide whenever it is exactly that an apple stops being an apple once we start eating it that could mean that the apple-ness of said apple has big chances of remaining intact (together with the underlying apple) irrespective of us eating it fully, and hence (theoretically) giving us an infinite supply of apples (or access to apple-ness, to be more exact).
But, going back to the Old Greeks, this is a very old question that we haven't been able to fully solve, I honestly think we'll never be able to solve it. Aristotle's focus on categories and especially his tertium non datur thing has allowed us to solve some practical problems (for example by allowing us to build a world that is based on techne, which among other things has allowed us to have this conversation here on the internet), but the bigger and more philosophical question of One vs. Infinity and everything in between remains un-solved.
Later edit: On Cantor vs. Aristotle, from here [1]
> Thus Cantor believed that Aristotle was quite mistaken in his analysis of the infinite, and that his authority was exceedingly detrimental
When it comes to the philosophy of mathematics and to philosophy itself I'm with Cantor on this. Could be that mainstream mathematics itself could benefit were we to ditch Aristotle and fully embrace Cantor, but I'm not a mathematician nor smart enough to say if that's in the realm of the possible.
[1] https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Canto...