Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.
"Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false."
It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?"
"No holes."
"Impossible!
"Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"
Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms."
"But we have the condition of continuity: We can keep on cutting!"
"No, you said an orange, so I assumed that you meant a real orange."
So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.
The surprising thing about BT is that the "pieces" are "moved around" ... there's no expansion or contraction.
Yes, the natural number thing helps to understand that simply counting things doesn't help, but the "rigid motion" aspect of BT takes it further.
And I agree that most people don't (initially) understand that an infinite set can be divided into two infinite sets that kinda "look the same", such as dividing Z (or N) into the evens and odds.
But BT is more than that. What follows isn't really for you, but is for anyone following the conversation.
Let's take a set A. It's a subset of the unit sphere, and it's a carefully chosen, special set, not just any random set. It's complicated to define, and requires the Axiom of Choice to do so, but that's what the BT theorem does ... it shows us how to define the set A.
One of the properties of A is that we can rotate it into a new position, r(A), where none of the points of r(A) are in the same position as any points of the original position, A. So the sets r(A) and A have a zero intersection. For the set A there are lots of possible choices of r ... we pick a specific one that has some special properties. Again, the BT theorem is all about showing us how to do this.
Now we take the union: B = A u r(A)
The bizarre thing is this. If we've chosen A and r (and therefore by implication, B) carefully enough, it ends up that there's another rotation, call it s, such that s(B)=A, the set we started with.
So whatever the volume of A, the volume of A u r(A) must be twice that, but that's B, and B can be rotated to give A back to us. So B must have the same volume as A. So 2 times V(A) must equal V(A), so A must have zero volume.
Well, we can kinda cope with that.
But if we've chosen A carefully enough, we find that a small, finite number of them, carefully chosen and rotated appropriately, together make up effectively the entire sphere (we miss out countably many points, but they have zero total volume, and we can fix that up later). So if finitely many copies of A make up a solid sphere, they can't have zero volume.
And that's the "paradox".
The conclusion is that we can't assign a concept of "volume" to the set A, and this is explained a little more in a blog post I've submitted here before:
https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo...
There's a lot more going on than just the "I can split infinite sets into multiple pieces that kinda look the same as the original", although that is certainly part of it, and lots of people already find that hard to take.
To any who has got this far, I hope that's useful.
>> ... the "two copies of the natural numbers" is sorta fine, except that they're more "spread out" ...
> What do you mean "spread out"? Aren't there the same amount of even numbers as natural numbers?
Yes, there are the same number, but when you look at just the even numbers, they are each distance 2 from their neighbours, whereas the natural numbers are all distance 1 from their neighbours. So people are less surprised, because the even numbers are "spread out", they are less dense in any given area. To map the even numbers back onto the natural numbers you have to "compress" them.
But this is not the case with the Banach-Tarski Theorem. There is a set, A, and another set B, which is just A rotated around, and they are disjoint. So they have a union, C=AuB. But when you rotate C, you can get an exact copy of A. There's no squashing or spreading needed.
So we have A and B, with B=r(A), and A intersect B is empty. Then we have C=AuB. No problem here.
The challenge comes that there is a rotation, s, such that s(C)=A.
So even though C is made up of two copies of A, it's actually identical to A. So start with C, divide it into A and B, then rotate B back to become a copy of A, and then rotate each of those to become copies of C. So you start with C, do some "cutting" and rotations, and you get two copies of C.
Finally, when you take a few of these and put them together, you get a full sphere, so you can't say they have zero volume.
Does that make sense? Does that answer your question?
Does that help?
The issue isn't the reals, but that "solid object" isn't defined properly, ie., the sets under question don't have well-defined volumes.
As soon as you fix that problem, via measure theory, the paradox resolves. You dont need to ditch real numbers.
There's really no evidence for this, as far as we know the real numbers are a pure mathematical invention and don't have any physicality.
Even if you want to say that spacetime is dense (i.e. infinitely divisible), there's an infinite number of fields like that, the real numbers are just a convenient superset.
There's no evidence that spacetime is dense either, and many practical ways in which it is not, as an obvious upper bound if you took all the energy in the observable universe to make one photon, it would still have a finite wavelength.
In a way the real numbers are a model (or maybe a 'language') to describe physical phenomena. They work exceptionally well at that, but they are not backed by evidence and do come with (theoretical) limitations.
This bachelor's thesis is a good starting point [1], search for 'finite precision physics' or 'intuitionistic math/physics'.
Eg., QM is only linear in infinitely-dimensional real-spaces, etc.
Essentially of a physics uses real spaces indispensably. There is no evidence whatsoever that this is dispensible; other than the fever dreams of discrete mathematicians.
By "wrong", I mean, we know they can't predict everything correctly. QM itself can't derive relativity. Relativity doesn't have QM in it, and break down at extremes like black holes. They're both very, very, very accurate in their domains, but physics knows that neither theory has the domain of "the entire universe". This is not a wild claim by an HN commenter, this is consensus in the physics world, just perhaps not phrased in the way you're used to.
It's possible the eventual Grand Unified Theory will still have continuous space at its bottom, but it's also entirely possible it won't. Loop quantum gravity doesn't. And personally I expect some sort of new hybrid between continuous and discrete based on physics history; whenever in the past we've had a similar situation where it couldn't be X for this reason, but it couldn't be the obvious Not-X for some other reason, it has turned out to be something that had a bit of both in them, but wasn't either of them.
I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces.
However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics!
This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians simply wish it were the case.
It is true that *maybe* (!) spacetime will turn out discrete, and likewise, Hilbert spaces, etc. -- and all continuous and infinite dimensional things will be discretised.
This however is a project without a single textbook. There is no such physics. There are no empirical predictions. There are no theories. This is a project within discrete mathematics.
Yes, they are, or more accurate, they're not right enough for you to confidently assert the structure of space time at scales below the Planck scale. You are doing so on the basis of theories known to be broken at that scale. You are not entitled to use the theories that way.
Even the Planck scale being the limit is a mathematical number; I'm not sure we have concrete evidence of that size being the limit. I've seen a few proposed experiments that would measure at that resolution (such as certain predictions made by LQG about light traveling very long distances and different wavelengths traveling at very slightly different speeds) but I'm not aware of any that have panned out enough to have a solid result of any kind.
The real numbers are popular outside of mathematical analysis because they provide a "kitchen sink" of every number you could possibly need.
The downside is that the reals include many numbers that you don't need. The number 0.12345678910111213... is a transcendental real number, but it is not very useful for anything. It is notoriously difficult to prove that a given number is transcendental, i.e. part of the uncountable part of the reals and not the countable algebraic subset. Which is ironic because the uncountable part is infinitely larger!
I'm not suggesting that physicists should drop their Hilbert spaces. Rather that a distinction should be drawn between mathematical model and physical reality.
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As for whether spacetime is countably infinitely divisible:
Infinity is big. Infinitely small implies that if you used all the atoms in the universe to write in scientific notation to write 10^-999..., that space would be more divisible than that. In fact for whatever absurdly tiny number you could think of, perhaps 1/(TREE iterated TREE(3) times) spacetime would be finer than that.
I'll admit it's possible, but I have trouble believing it.
I don't see that these properties are incidental.
Yes they obtain in virtue of /any possible "dividing" discrete sequential process/ never terminating, eg., space being "infinitely divisible".
However I dont think this is as bizarre as it appears. The issue is congition is discrete, but the world continuous.
So we are always trying to project discrete sequential processes out onto the world in order to reason about it. Iterated zooming-in will, indeed, never terminate.
I dont see that as saying anything more than continuity produces infinities when approached discretely. So, don't approach it that way, if that bothers you.
But if we accept that the Planck length is the smallest possible length and the Planck time is the smallest possible time, then it seems logical that the universe is an integer lattice of these. ("Spacetime is not real-valued")
https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_time
However the idea that space time is discrete is a reasonable hypothesis to test, we don’t currently have any ways to probe at those resolutions, though.