Well, I have two apples on my desk. I can't have an infinity of anything. In that sense 2 exists more tangibly than infinity.
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 see, so the belief is that improved theories of physics will remove all the infinities we appear to see around us?
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.
> will remove all the infinities we appear to see around us?
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.
One example would be the infinite number of ways there are for humans to form unsound conclusions. But of course, this is not "real" reality that can be tidily corralled with math/science so it doesn't count, but it's still interesting.
According to the Big Bang model the universe started with a super small volume. When did it get to infinity?
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.
This is a common misconception, in standard models of big-bang cosmology the universe was always infinite. It was hotter and denser earlier, and is cooler and less dense later but always 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.
That's not how physics works. We can't prove anything. All we do is come up with worse or better models, and the best model anyone has come up with so far has the universe being infinite. Finite universe models have some problems, since the global curvature of the universe is (as best we can measure) zero the universe is either flat or pretending to be flat. Finite volume flat spaces can be characterised and they're a bit weird (1).
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.
I don't say it has to be isotropic, but observationally it looks very close to isotropic. There are quite clear signals we'd expect to see in the cosmic microwave background if the universe was anisotropic and they aren't there:
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.
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.
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.
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.
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.
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.
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?
Are you sure you have two apples on your desk? What happens if you take a small bite? Is it still two apples? When will said apple "cease" to exist as an unit while you're eating it?
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.
I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. At some point some will start disagreeing, but with two big full apples, all of them will agree.
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.
> I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition.
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)
Going by my (bad) analogy, in theory, yes, that is what that would mean.
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.
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"