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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#51

Post by hgsgm »

karmakaze wrote: post #12

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 radi…

The difference between [0, 1] and B-T's shapes is that B-T's shapes have uncountably-infinite complexity of details. This is what constructivists and finitists and "constructivism + excluded middle"-ists do not accept.

https://ncatlab.org/nlab/show/constructive+mathematics

Re: The Point of the Banach-Tarski Theorem

#52

Post by hgsgm »

josephcsible wrote: post #42
ur-whale wrote: post #41

So, Banach-Tarski says you can split a sphere of volume S into a finite number of pieces and reassemble the pieces into two spheres of volume S. Is the proof constructive? As in: does the proof actually show how to build the pieces? If yes, is the boundary of the pieces of measurable surface? Can the pieces be rendered in 3D? Or is it just another one of those "proofs" where if the set of pieces doesn't exist we land…

https://youtu.be/s86-Z-CbaHA?t=673 nicely visualizes it. It's an infinite number of infinitely thin filaments from the center of the sphere going outward in every direction.

Not just infinite, uncountably infinite!

Re: The Point of the Banach-Tarski Theorem

#53

Post by hgsgm »

CJefferson wrote: post #2

I always feel part of the confusion with Banach-Tarski is that lots of words don't use their "natural definitions", which makes the proof more surprising. People (not this article) often talking about "cutting" a sphere, which is really misleading. This result is, in many ways, quite similar to the idea I can "cut" the integers into the odd integers and even integers (but with many more fine details). This is still a…

> (but with many more fine details).

Uncountably many more fine details.

Re: The Point of the Banach-Tarski Theorem

#54

Post by fellowmartian »

camjw wrote: post #25
moloch-hai wrote: post #20

Earlier quoted context omitted.

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.

It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?

in my view they never were "numbers" (as in constants), they are functions

Re: The Point of the Banach-Tarski Theorem

#56

Post by thaumasiotes »

jiggawatts wrote: post #49
thaumasiotes wrote: post #45

Earlier quoted context omitted.

> Actually, an ever better example is: "The set of reals that are not the solution to any equation that can be written with a finite number of symbols." -- an infinite set that has no nameable members! That's not a good example; the problem you're creating is due to sloppy use of language, not any cleverness in the definition. All real numbers, and all numbers of any other variety, can be written with a finite number…

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counter-intuitively, this is not true. The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number…

> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long.

So what? That doesn't stop them from being named.

Re: The Point of the Banach-Tarski Theorem

#57

Post by [deleted] »

joerichey wrote: post #6
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."

If you accept that _all_ vector spaces have a (Hamel) basis, you can then prove the Axiom of Choice: http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf This means if you want to deny the Axiom in some cases, you will also have to allow for the existence of vector spaces without a basis.

[deleted]

Re: The Point of the Banach-Tarski Theorem

#58

Post by drdec »

moloch-hai wrote: post #29
ttctciyf wrote: post #24

Earlier quoted context omitted.

> Wherever it comes down to actual numbers Wait, are you saying e and pi aren't "actual numbers"?

I am not a Platonist, or any sort of theist. e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world. I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no bet…

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature.

Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.

Re: The Point of the Banach-Tarski Theorem

#59

Post by drdec »

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

Keep in mind that "pieces" might not mean the same thing you think of it as meaning.

Re: The Point of the Banach-Tarski Theorem

#60

Post by moloch-hai »

drdec wrote: post #58
moloch-hai wrote: post #29

Earlier quoted context omitted.

I am not a Platonist, or any sort of theist. e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world. I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no bet…

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.

I probably confused matters with my undefined expression "actual numbers". e and pi are as actual as i or 0.

I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their existence is not just countable, but finite and not very large. You have to have expressed a representation of each in it at least once. We can only express a small number of numbers over the course of a life. Computers can do more of them. You might claim all those that programs you wrote have expressed.

And, of course, the number has to have finite expression. Representing pi as a summation is finite. But presenting a numerical computation involving it has to be an approximation if it ever to finish.

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