The diagonal proof is just arguing that two nested while(true) loops will run for longer than one. (And then we define this as "bigger" just to confuse undergraduates)
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...
If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.
So be it, but it is a useless definition of exists then isn't it?
If it makes you feel better to put infinity into the same category of reality as "unicorns", then I think that's fine? It doesn't necessarily need to be the case that anything in mathematics corresponds to something that is physically manifested, and probably _most_ of mathematics does not -- not just infinities.
Infinities are quite different, given that it is by definition impossible to measure whether something in reality is infinite or not unless you have infinite time.
It's impossible to count a negative number of things. It's impossible to count an imaginary number of things, it's impossible to count a matrix of things. It's impossible to count an irrational number of things. These are all fictions that people use because they're useful. Infinity is exactly the same as any of those other concepts.
Also I clearly just demonstrated that you can count negative numbers easily so I don't understand why you find it impossible.
The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.
Neither. That's my point. Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state. To say that one of these functions can produce more distinct states than another is pointless, because the procedure that produces the most states will always be the one that you ran the most times.
There is nothing observably infinite, since it would take infinite time to observe that any given thing was infinite. The only possible proof of infinity would be a machine that runs infinitely quickly. e.g. https://qntm.org/responsibility
This combination of words seems strange.
Like: proof of ‘zero’ or proof of ‘left’.
All of them have definitions, not proofs.
(Qualitative distinction of different infinity types has a proof though)
Could you come up with or point to such a procedure for R (the reals)?
As I understand the diagonalization argument you can do that for N, but not for R.
This will never reach 2, so it will not generate all real numbers. (Which was what parent was asking for, to recusively generate all R)
This is how natural numbers work in the first place. You're just adding a decimal point to all of the possible places it could go.
When will this reach PI or e or sqrt(2)?
(There are infinitely many numbers that will not be reached by this procedure)
How is this? This is because PI is not actually a number. It's a procedure that generates digits for approximating things about circles.
This is the same with sqrt(2). The sqrt procedure emits digits just as the procedure to find all real numbers does.
You can't "reach" PI for the same reason that a natural number can't reach "f(x) => x + 1". That is, natural numbers aren't procedures or functions.
What about 1/3, 1/7, …? Previously outlines recursive procedure doesn’t generate those.
But yeah, if you deny existance of irrational numbers, and redefine Real:=Rational, then you can generate these “real” numbers recursively and it does follow that all infinities have same cardinality here.
Btw. what is the diagonal of a unit square formed by 4 objects at the corners? I assume it is a rational number. Btw2. If you take that answer and multiply by itself, what do you get?
Important to note: When ggp asked for a recursive procedure to generate real numbers, they wanted that exactly same proceedure would generate all reals (not special procedure for each number)
If we have special procedure for each number, then procedure to generate 1/3 is just 1/3. …of course naively assuming notation of 1/3 is as valid as 0.33333…, and that base 10 is not the only possible base.
1/3 isn't a real, it's a fraction. Fractions can be used to generate reals, and they can be used in algebra along with reals.
0.(3) is also not a real. It's also just representative of a procedure that can generate reals.
Both 1/3 and 0.(3) can still be used in algebra in the same way as before. You don't lose any capability because you can't practically expand 0.(3) to infinite decimal places in the first place.
What about same number expressed in base 3? (I think in base 3 that would be written as 1/31)
And what about number 0.1 in base 3? (Which is equivalent to 1/3 in base 10)
Does it really have an infinite representation? I can't imagine an infinite representation fitting on a page. I'm pretty sure you're representing it as 1/3 or 0.(3). Neither of those representations are infinite. They're only a few characters really.
Are these numbers the same: “0.5 in base10” and “0.1 in base2”?
1 => first step in the procedure 2 => second step ...
That in turn would mean that N and R have the same cardinality. This would be news.
But in all probability we are discussing the wrong thing here. Our difference it's likely at a deeper conceptual level than this.
You can’t generate R this way. This is a consequence of Cantor’s proof.
Also, what has nested loops got to do with it? You can use one loop to generate the natural numbers, and a pair of nested loops to generate pairs of natural numbers (if you like, the rationals). But the diagonal proof doesn’t show that these will have ‘different’ run times — they’re in fact in bijective correspondence.
You can come up with a bunch of procedures that generate numbers indefinitely, and you can even define relationships between those procedures, but a procedure is not a number.