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Banach-Tarski and the Paradox of Infinite Cloning

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Re: Banach-Tarski and the Paradox of Infinite Cloning

#81

Post by karmakaze »

ishtanbul wrote: post #61

Vsauce made a great video about this https://youtu.be/s86-Z-CbaHA

I recall watching the video and not being surprised by its paradox. The set of starting points is uncountably infinite (R2), and since each starting point leads to a countably infinite number of L/R/U/D-rotation-ending sets, each of those L/R/U/D sets has the same cardinality as that for starting points. And so on. In the end, what I took away from this was similar to saying the interval [0.0, 0.5] has the same cardinality as [0.0, 1.0] albeit in a higher number of dimensions. It would be surprising if an uncountably infinite set in a lower dimension could fill in a higher one, but uncountably infinities in the same number of dimensions doesn't seem like a paradox that needs this sphere, rotation, and dictionaries to demonstrate.

In reading the comments for the video, I got the sense that this is different and that I was missing something but couldn't come close to guessing what that was.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#82

Post by pfortuny »

thaumasiotes wrote: post #39
Sinidir wrote: post #27

Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…

> The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume). > This seems incorrect. It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following: 1.…

“Cutting” it is certainly not: none of those sets is given by the zeroes of a continuous function (they would be measurable, and they cannot be).

So the paradox breaks down when you start to realize that you are not CUTTING but “choosing some points” and rearranging them. The fact that this rearrangement can be done with Euclidean moves is the surprise.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#83

Post by [deleted] »

thaumasiotes wrote: post #70
ncallaway wrote: post #64

Earlier quoted context omitted.

> The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axio…

> But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere. The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist. But…

[deleted]

Re: Banach-Tarski and the Paradox of Infinite Cloning

#84

Post by layer8 »

danparsonson wrote: post #34
jhgb wrote: post #22

Earlier quoted context omitted.

> infinity is something only present in our math and not in the universe This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.

OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.

But the electron field has different values at different points in spacetime, and we have no evidence that either the number of points (locations) or the number of different possible values at those points is finite. Unless we posit that they are finite in number, infinity is quite present in the universe.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#85

Post by thaumasiotes »

mjw1007 wrote: post #76
thaumasiotes wrote: post #70

Earlier quoted context omitted.

> But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere. The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist. But…

I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number". But the Twitter link at the top of this thread seems to have a rather more interesting way of doing so.

> I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number".

The Banach-Tarski theorem is not the only theorem out there that bothers some people. Anything to do with infinities gets a large number of outraged rejections.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#86

Post by snet0 »

tsimionescu wrote: post #55
snet0 wrote: post #41

Earlier quoted context omitted.

That's just because you use the word "exact", though. Exactitude doesn't exist in the universe as we understand it. There's a difference between something not being instantiated in this universe and being unscientific, though. If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it u…

> Exactitude doesn't exist in the universe as we understand it. Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge. > There's a difference between something not being instantiated in this universe and being unscientific, though. Well, science is a particular way of studying what exists. Studying something that doesn't exist is unscientific (of course, y…

You were talking about exactitude in space, rather than charge.

>Studying something that doesn't exist is unscientific

What about things that could exist, might exist, or even aren't expressly forbidden from existing? These have all been used as perfectly valid reasons for scientific inquiry, historically.

Asking "did some god create the universe" is unscientific by your reasoning so long as it is known that there is no in-universe trace or evidence that it was indeed created by a god. Proving that is proving a negative. I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something. It does certainly feel impossible that we will prove that the universe wasn't created by a god, though. The inquiry is deemed unscientific because we have no reason to go down that pathway, not because the question is fundamentally intractable.

Multiverse theory, on the other hand, would qualify as unscientific by your reasoning. If it were true, the different universes would be fundamentally inaccessible, according to our understanding. The model does not suggest that evidence could even possibly exist, as far as I understand.

A result being untestable doesn't, in my opinion, lead to it being unscientific. We cannot test whether black holes exist, except by looking for them. We cannot test whether wormholes exist, except by looking for them. These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result, just because maybe it is the case that our model is correct but there was never the appropriate state of the universe to prove our prediction.

Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct? Is that unscientific? Just like we assume that the charge on electrons is constant despite not actually measuring it always everywhere.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#87

Post by karmakaze »

throwaway81523 wrote: post #59
thaumasiotes wrote: post #45

Earlier quoted context omitted.

Come to think of it, the fact that two spheres contain the same number of points as one sphere does would seem to be closely related to why it's possible to produce two spheres from one sphere just by rearranging the points. You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an unc…

It is deeper than that. There is no way to do a similar duplication of a 2-dimensional disc. Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

This comment and the parent sheds some light on what makes this interesting. It didn't occur to me that we were allowing rotations and translations and that scaling is excluded. The paradox isn't about getting more from less as they're all comparably infinite, but rather being able to arrange them to be so.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#88

Post by thaumasiotes »

throwaway81523 wrote: post #59
thaumasiotes wrote: post #45

Earlier quoted context omitted.

Come to think of it, the fact that two spheres contain the same number of points as one sphere does would seem to be closely related to why it's possible to produce two spheres from one sphere just by rearranging the points. You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an unc…

It is deeper than that. There is no way to do a similar duplication of a 2-dimensional disc. Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

> Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

Can you be more specific? Rotations and translations also exist in 2-space. It seems difficult to argue that this difference between 2-space and 3-space is "not a property of 3-space".

Re: Banach-Tarski and the Paradox of Infinite Cloning

#89

Post by [deleted] »

PeterisP wrote: post #53
jhgb wrote: post #49

Earlier quoted context omitted.

That sounds like a weird interpretation of "to be present in the universe" to me. Also I was under the impression that it's unknown whether the universe contains an infinite number of electrons or not.

It's certainly known that the observable universe does not contain an infinite number of electrons, as it has a finite size and finite mass. And it's rather moot to talk about the space beyond the observable universe that can never affect us or anything we can observe in any way whatsoever, so any other statements about it are inherently unfalsifiable, so all the science of physics is relevant only w.r.t. the (finite…

[deleted]

Re: Banach-Tarski and the Paradox of Infinite Cloning

#90

Post by ithkuil »

tsimionescu wrote: post #79
ithkuil wrote: post #67

Earlier quoted context omitted.

Infinity may be also "necessary to describe the world". But like every tool, you need to know its limits.

I'm not sure that it could be, actually. You can't use a finite amount of evidence to verify that something is infinite, so any infinity can always be replaced with a huge (or minuscule) number and the theory would make the same measurable predictions.

I'm not talking about proving that there exist infinite things.

I'm talking about using the abstract concepts of infinity as a useful mathematical tool to produce predictions. Notable example: calculus

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