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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#11

Post by zokier »

moloch-hai wrote: post #5

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…

> It relies on uncountably-infinite division of an object

But the theorem claims finite division and not infinite?

In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R

Re: The Point of the Banach-Tarski Theorem

#12

Post by karmakaze »

I'm glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn't get the significance of it. It either didn't seem like an unexpected result or too esoteric to appreciate.

There's phrasing in the post that could be misunderstood (later clarified) but can leave unclarity from assumed understanding of the earlier description.

> In R3, given a solid ball B of radius R, it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R.

"finitely many pieces" could be confusing because though we may be talking about 6 'pieces' those pieces have an uncountable infinity of radial line slices. It's also missing the rigid motion part which is key.

The part that didn't seem surprising (and is considered trivial) was in handling uncountable infinities of things. The 'number' of points on a line [0, 1) is the same as the number on a line [0, 2) so I wouldn't be surprised to map points from [0, 1) to [0, 2) filling the latter without 'gaps'. Similarly for areas. But what the theorem is saying is that this kind of mapping doesn't work in R1 nor R2 but does work in R3 (with rigid motions).

The part that makes B-T surprising is that the extra volume/ball can be constructed with rigid motions of those uncountably infinite sets. This is where it seems beyond me to appreciate: that one or two balls have the same uncountably number of radial line slices is considered trivial, and mapping using rigid motions is surprising.

For example, if you do the same rearrangement but instead of sets of radial lines, consider the set of points at the surface of those radial lines, we're basically working in R2. The reason why it can't be done is because if we instead of being on the surface of a sphere we're on a plane, then those same motions aren't distance preserving. Thinking geometrically doesn't seem weird: an extra dimension let's you do something you can't in lower ones.

I think the algebraic description that the post makes may be clearer to appreciate the difference, and I'll be giving it another read and more thought.

[0] https://www.youtube.com/watch?v=s86-Z-CbaHA

Re: The Point of the Banach-Tarski Theorem

#13

Post by moloch-hai »

Aicy wrote: post #9
moloch-hai wrote: post #5

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…

In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete. We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing? > we use floating-point numbers, not reals, when doing actual calculations I don't believe only calculations made by computers to…

No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.

Re: The Point of the Banach-Tarski Theorem

#14

Post by moloch-hai »

zokier wrote: post #11
moloch-hai wrote: post #5

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…

> It relies on uncountably-infinite division of an object But the theorem claims finite division and not infinite? In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R

Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.

Re: The Point of the Banach-Tarski Theorem

#15

Post by ttctciyf »

moloch-hai wrote: post #13
Aicy wrote: post #9

Earlier quoted context omitted.

In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete. We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing? > we use floating-point numbers, not reals, when doing actual calculations I don't believe only calculations made by computers to…

No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

Re: The Point of the Banach-Tarski Theorem

#16

Post by [deleted] »

Aicy wrote: post #9
moloch-hai wrote: post #5

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…

In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete. We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing? > we use floating-point numbers, not reals, when doing actual calculations I don't believe only calculations made by computers to…

[deleted]

Re: The Point of the Banach-Tarski Theorem

#17

Post by Someone »

ttctciyf wrote: post #15
moloch-hai wrote: post #13

Earlier quoted context omitted.

No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

I don’t think so. Programs that use numbers approximate e and π. It’s only programs that manipulate symbols that can use their exact values.

Re: The Point of the Banach-Tarski Theorem

#18

Post by QuesnayJr »

whatshisface wrote: post #4

Here is a potentially daft question that I nonetheless would appreciate if someone could answer. Is it possible to deny the axiom of choice for the purposes of measures while accepting it for vector spaces? I am wondering if you could say, "there are two kinds of sets, ones equipped with a choice function and ones without it, and measurable sets are of the latter kind."

You probably could, though you would probably have to give up or weaken other axioms as well. You would need to prevent there being any bijections between the good sets and the bad sets, because you then you could use the bijection to define a choice function on the bad set. What set theorists usually do is just distinguish between when you use the axiom of choice and when you don't.

There's one super-subtle point, though: even if you ban using the axiom of choice, you can still construct sets where it is independent of the axioms of set theory whether they are measurable. You have to add "these sets are measurable" as an axiom.

The simplest way to do it is to consider a class of sets where you know you didn't use the axiom of choice, and then declare that these sets are measurable. One common choice is the class of "projective sets". Usually set theorists impose a stronger property called determinacy, so they add to set theory the Axiom of Projective Determinacy. Then you can say "If I stick to the good sets -- projective sets -- nothing bad happens. Arbitrary choice functions take me out of the projective sets, where bad things can happen."

Re: The Point of the Banach-Tarski Theorem

#19

Post by [deleted] »

moloch-hai wrote: post #5

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…

[deleted]

Re: The Point of the Banach-Tarski Theorem

#20

Post by moloch-hai »

ttctciyf wrote: post #15
moloch-hai wrote: post #13

Earlier quoted context omitted.

No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.
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