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Banach-Tarski and the Paradox of Infinite Cloning

quantamagazine.org

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Re: Banach-Tarski and the Paradox of Infinite Cloning

#71

Post by auggierose »

tsimionescu wrote: post #12
turminal wrote: post #7

Earlier quoted context omitted.

> there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time That depends on the model of computation you pick, doesn't it?

Only if you want to think about models of computation that allow performing infinite operations in finite amounts of time, which I don't think are that interesting.

I'd say if our universe makes such computations possible, then that would be very interesting.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#72

Post by thaumasiotes »

joiguru wrote: post #68
thaumasiotes wrote: post #39

Earlier quoted context omitted.

> 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.…

"cut the ball into 5 pieces" is not the best description. A better one is: 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

> 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

Huh? The ball is already composed of infinite points. So in 2a you recognize that the ball exists, and then in 2b you cut it into pieces. But it seems superfluous to mention 2a separately.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#73

Post by joiguru »

thaumasiotes wrote: post #72
joiguru wrote: post #68

Earlier quoted context omitted.

"cut the ball into 5 pieces" is not the best description. A better one is: 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

> 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups Huh? The ball is already composed of infinite points. So in 2a you recognize that the ball exists, and then in 2b you cut it into pieces. But it seems superfluous to mention 2a separately.

In regular natural language, cutting the ball into 5 pieces implies cutting 5 contiguous pieces.

If I say a cake is cut into 5 pieces, no person will consider that each piece contains parts from all parts of the cake.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#74

Post by KirillPanov »

Spoiler: the cut line between the two apple-halves is fractal in shape, with infinite surface area and taking forever to cut at any finite cutting speed.

It's sort of hilarious to see a physics site mention the Banach-Tarski paradox. It is, after all, the most obvious hole poked in the most basic working assumption used by physicists: that space and time are measured with real numbers.

I've seen physicists go to pretty absurd extremes to avoid thinking about the problems this creates. Fixing it properly is not easy: simply dropping the axiom of choice leaves you unable to do useful physics. Getting back to a useful state, making all sets Lebesgue, can only be done with large cardinals:

https://www.jstor.org/stable/1970696

Large cardinals are pretty exotic even by the standards of mathematicians. In many departments they are in fact the domain of logicians. In fact, the existence of certain classes of Woodin cardinals is equivalent to the Axiom of Determinacy (AD), which is the "mathematically respectable" way of investigating logics with infinitary conjunction/disjunction. In fact, AD is precisely the Law of Excluded Middle (A or not-A) for logics with infinitely-long conjunctions.

Quite odd that something so ethereal would be connected to a tangible act like cutting an apple in half.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#75

Post by tsimionescu »

auggierose wrote: post #71
tsimionescu wrote: post #12

Earlier quoted context omitted.

Only if you want to think about models of computation that allow performing infinite operations in finite amounts of time, which I don't think are that interesting.

I'd say if our universe makes such computations possible, then that would be very interesting.

Absolutely! Similarly, if our universe allowed instantaneous travel and free energy, that would also be very interesting.

Not holding my breath for either.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#76

Post by mjw1007 »

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…

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.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#77

Post by tsimionescu »

canjobear wrote: post #62
tsimionescu wrote: post #55

Earlier quoted context omitted.

> 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…

> Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge. Aren't claims like this unscientific according to your standard? You will never be able to measure that two electrons have the same charge to infinite decimal precision. You might have a theory that says they should have the same charge, but you won't be able to test that theory to infinite precision…

Hmm, actually you may be right. I'm not entirely sure how powerful the 'requirement' in QM for these quantities to be quantized is though, but most likely you are right - the theory wouldn't be able to distinguish between identical charges and veeeeery slightly different charges.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#78

Post by pfortuny »

mjw1007 wrote: post #50
thaumasiotes wrote: post #47

Earlier quoted context omitted.

> I remember sitting in maths lectures and wishing that when they did thing like prove the intermediate value theorem they'd make it clearer that what was going on wasn't so much "We're rigorously proving that this thing that seems obvious is true" as "We're checking that the formalisation we introduced earlier is fit for purpose". > I think things like the Banach-Tarski theorem are the other side of that coin: they'…

What I mean is: if you imagine someone drawing up a requirements document for the team assigned to the task of axiomatising geometry, and somebody asked "Do we want our model of geometry to support cutting up a ball into five pieces, moving the pieces rigidly, and reassembling them into two copies?", I think their first idea would be to answer "no". So it isn't parallel to the intermediate value theorem, but opposite…

The thingis exactly that in the B-T paradox you are NOT “cutting”, you are “choosing” non-measurable subsets and reorganizing them. Thus, the operation of “cutting” is not taking place (there is no continuous function whose zeros gives you any of the subsets).

Re: Banach-Tarski and the Paradox of Infinite Cloning

#79

Post by tsimionescu »

ithkuil wrote: post #67
tsimionescu wrote: post #31

Earlier quoted context omitted.

Whole numbers can be defined and proven to be necessary to describe the world pretty easily. From there, rational numbers are trivial to define. Negative numbers are somewhat more abstract, but they have very intuitive definitions in many domains, such as accounting. It may be possible to avoid them in a theory of physics, though. The complex numbers (well, at least those with a rational imaginary part and a rational…

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.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#80

Post by Ericson2314 »

thaumasiotes wrote: post #57
mjw1007 wrote: post #50

Earlier quoted context omitted.

What I mean is: if you imagine someone drawing up a requirements document for the team assigned to the task of axiomatising geometry, and somebody asked "Do we want our model of geometry to support cutting up a ball into five pieces, moving the pieces rigidly, and reassembling them into two copies?", I think their first idea would be to answer "no". So it isn't parallel to the intermediate value theorem, but opposite…

The whole idea of a proof system is that there are some things you can't have without also having other things. The Banach-Tarski theorem is a consequence of things we want. You don't get to pick and choose everything at once.

Did you read the original link? We don't want to use topological spaces, we want to use locales! We can get rid of the paradox while sacrificing very little.
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