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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#91

Post by [deleted] »

theemathas wrote: post #47
jiggawatts wrote: post #33

Earlier quoted context omitted.

The Axiom of choice has never felt completely self-evident to me. E.g.: what if you have sets where the elements are non-computable? How do you "choose" objects that cannot even be named ? Something like: "the set of all programs that cannot be proven to halt" and the like can be used to create pathological sets where the set itself obviously exists, but you cannot name any of the members. Actually, an ever better ex…

An alternative formulation of the axiom of choice: The cartesian product of a collection of non-empty sets is non-empty.

[deleted]

Re: The Point of the Banach-Tarski Theorem

#92

Post by ColinWright »

SaintRomuald wrote: post #88

I never understood this example used to explain it, vsauce made a video on the banach tarski theorem. You make an infinite list of numbers between 0 and 1 chosen at random. Apparently you can make a new number that was never seen in the list before if you pick a digit from each number in the list and add one to it. Say the list has numbers 0.36285728.. 0.95825597.. 0.47264112.. .. I can make a new number by taking th…

You have misunderstood, or misremembered.

You don't take "3" from the first, you consider three, but choose something that's not "3", so the number you are constructing differs from that first number.

Then in the second place you don't take "5", you choose something that's not "5", so the number you are constructing differs from that second number.

And so on. So every time you have a list of real numbers, it cannot contain all real numbers ... you can always construct many, many, many numbers that are not in your list.

But I'm not sure how this is related to what we've been talking about. If you understand what I've said here and are still confused, maybe you can be a bit more specific. If you have not understood what I have said here, perhaps you can ask more specific questions.

Re: The Point of the Banach-Tarski Theorem

#93

Post by WastingMyTime89 »

thaumasiotes wrote: post #45
jiggawatts wrote: post #33

Earlier quoted context omitted.

The Axiom of choice has never felt completely self-evident to me. E.g.: what if you have sets where the elements are non-computable? How do you "choose" objects that cannot even be named ? Something like: "the set of all programs that cannot be proven to halt" and the like can be used to create pathological sets where the set itself obviously exists, but you cannot name any of the members. Actually, an ever better ex…

> 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.

This is untrue and not particularly hard to prove by contradiction.

Suppose every elements in R can be named by a finite number of symbols.

You can build a bijection between R and the names of its elements (by definition a name points to a unique element and if elements have multiple names, you can easily well order a finite number symbols by building a lexicographic order and only consider the smallest name).

Or, you can also easily build a bijection between a finite number of symbols and N. That's just an encoding like ASCII or UTF-8. Therefore, the set of names is countably infinite.

Yet R is uncountable (see Cantor's diagonal argument).

Therefore, by contradiction, there has to exist elements in R which can't be named by a finite number of symbols.

Re: The Point of the Banach-Tarski Theorem

#94

Post by moloch-hai »

KvanteKat wrote: post #87
moloch-hai wrote: post #60

Earlier quoted context omitted.

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…

How is "The circumferance of an idealized circle divided by its diameter" not a finite expression of π? Saying something cannot be expressed finitely in an integer-based numeral system, and saying that it admits no finite representation are two radically different statements. Despite it being a non-starter from a pragmatic standpoint, we could for instance easily imagine a novel numeral type that encodes the set S =…

> Despite it being a non-starter from a pragmatic standpoint

That was my point, thanks.

Re: The Point of the Banach-Tarski Theorem

#95

Post by quchen »

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.

It’s like that for all numbers, not just the fancier ones! I have never experienced a 1.

Re: The Point of the Banach-Tarski Theorem

#96

Post by CJefferson »

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…

Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity.

I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)

Re: The Point of the Banach-Tarski Theorem

#97

Post by yellowapple »

smcl wrote: post #80
yellowapple wrote: post #34

"So for those of you who don't know the result, here it is in simple, non-technical terms:" proceeds to immediately use a character that can't even be copy/pasted due to needing MathJax to render it, refuses to elaborate further Thankfully, it ain't hard to find the Banach-Tarski theorem in actual simple, non-technical terms, so for those wondering what in tarnation that character is: it just means three-dimensional…

I did get a chuckle at that, like the running joke about how a monad is “just a monoid in the category of endofunctors”. To be fair, you could delete the “R^3” part and the explanation can still be understood.

Hell, deleting the 𝕽³ part probably would've made it clearer: "ball" already implies 3D space for me, whereas seeing that qualified with some janky-ass medieval R had me second-guessing about it being one of those math quirks involving non-euclidean space or imaginary-number dimensions or whatever else mathematicians come up with in their efforts to compute the passcode on the Gates of Hell.

Re: The Point of the Banach-Tarski Theorem

#98

Post by karmakaze »

hgsgm wrote: post #51
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

That the operation would work on the '2D' surface of a sphere but not when planar suggests that the complexity isn't the issue, it's the geometry.

Re: The Point of the Banach-Tarski Theorem

#99

Post by smcl »

yellowapple wrote: post #97
smcl wrote: post #80

Earlier quoted context omitted.

I did get a chuckle at that, like the running joke about how a monad is “just a monoid in the category of endofunctors”. To be fair, you could delete the “R^3” part and the explanation can still be understood.

Hell, deleting the 𝕽³ part probably would've made it clearer: "ball" already implies 3D space for me, whereas seeing that qualified with some janky-ass medieval R had me second-guessing about it being one of those math quirks involving non-euclidean space or imaginary-number dimensions or whatever else mathematicians come up with in their efforts to compute the passcode on the Gates of Hell.

Yeah I feel you - I think these things come so naturally to mathematicians that it's hard for them to conceive of the idea that people like you and I exist :)

Re: The Point of the Banach-Tarski Theorem

#100

Post by lodi »

CJefferson wrote: post #96
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…

Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity. I realise that's all very dodgy, but of course we are also b…

My understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.
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