Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Point of the Banach-Tarski Theorem

solipsys.co.uk

Post reply on HN
Print view View original

31–40 of 116 posts

Re: The Point of the Banach-Tarski Theorem

#31

Post by moloch-hai »

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?

They are not numbers you have ever used in an actual calculation. You always approximate them anyplace they don't cancel out.

There is a countable subset of the reals called "computable numbers". We only ever use a subset of computable numbers in real calculations.

Re: The Point of the Banach-Tarski Theorem

#32

Post by moloch-hai »

defrost wrote: post #27
moloch-hai wrote: post #22

Earlier quoted context omitted.

You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them. It is magic of a kind by wizards of a kind, or anyway indistinguishable…

Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end"). It's another of those intuition challenging moments in math.

It's the wackiest result I know of, and seems overwhelmingly more profound than BT.

Re: The Point of the Banach-Tarski Theorem

#33

Post by jiggawatts »

azeemba wrote: post #3

As the author points out that Banach-Tarski theorem is an example of hard-to-accept result that comes out of the easy-to-accept axiom of choice. There is a popular quote that related to this: > The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? From https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice, the well-ordering principle and Zorn'…

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 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!

Re: The Point of the Banach-Tarski Theorem

#34

Post by yellowapple »

"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 space (and is usually rendered as ℝ³ instead of 𝕽³, at least so Wikipedia claims).

Re: The Point of the Banach-Tarski Theorem

#35

Post by yellowapple »

gerdesj wrote: post #28
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…

It is a remarkable theorem but you can't take a snooker ball and turn it into two snooker balls without being Paul Daniels (UK magician). This is an excellent example of language going astray and confusing mathematic rigour with some sort of "reality". You can loosely model a "sphere in R3" with a snooker ball in errr the universe thingie which is probably R3ish or perhaps R3T1 or whatevs. Besides someone has spilt a…

Maybe Paul Daniels is just really really good at math?

Re: The Point of the Banach-Tarski Theorem

#36

Post by H8crilA »

jiggawatts wrote: post #33
azeemba wrote: post #3

As the author points out that Banach-Tarski theorem is an example of hard-to-accept result that comes out of the easy-to-accept axiom of choice. There is a popular quote that related to this: > The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? From https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice, the well-ordering principle and Zorn'…

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…

I think your second set doesn't exist (exactly like the set of all sets doesn't exist) under the axiom of choice, for I can use the axiom of choice to select an element, hence giving it a name :). I took some liberty with the allowable names, of course.

You still have a point, though.

Re: The Point of the Banach-Tarski Theorem

#37

Post by soVeryTired »

jiggawatts wrote: post #33
azeemba wrote: post #3

As the author points out that Banach-Tarski theorem is an example of hard-to-accept result that comes out of the easy-to-accept axiom of choice. There is a popular quote that related to this: > The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? From https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice, the well-ordering principle and Zorn'…

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…

Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).

The weirdness of the axiom of choice only really comes through when you consider bigger and bigger collections of sets.

For example:

- sets indexed by the natural numbers: S_1, S_2, ... It seems totally reasonable that you should be able to make a new set by picking something from the first set, something from the second set, etc

- sets indexed by continuous time (i.e. real numbers). Here it's a bit less 'obvious'. If I have sets S_t for _every_ time t > 0, can I really make choices 'fast' enough? What if the sets are so unstructured that I'm forced to stop and look at each set in turn to make my choice?

- sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?

I feel like the last example demonstrates how powerful the full axiom of choice actually is.

NB - I'm a dilettante rather than an actual logician, so there may be mathematical inaccuracies here.

Re: The Point of the Banach-Tarski Theorem

#38

Post by User23 »

yellowapple wrote: post #35
gerdesj wrote: post #28

Earlier quoted context omitted.

It is a remarkable theorem but you can't take a snooker ball and turn it into two snooker balls without being Paul Daniels (UK magician). This is an excellent example of language going astray and confusing mathematic rigour with some sort of "reality". You can loosely model a "sphere in R3" with a snooker ball in errr the universe thingie which is probably R3ish or perhaps R3T1 or whatevs. Besides someone has spilt a…

Maybe Paul Daniels is just really really good at math?

Rumor has it that Sir Isaac was investigating a mathematical praxis along such lines. For whatever reason the college isn’t willing to release those papers.

Re: The Point of the Banach-Tarski Theorem

#39

Post by lmm »

soVeryTired wrote: post #37
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…

Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible). The weirdness of the axiom of choice only really comes through when you consider bigger and bigger collections of sets. For example: - sets indexed by the natural numbers: S_1, S_2, ... It seems totally reasonable that you should be able to make a new set by picking something from the fi…

> Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).

You could form e.g. the non-algebraic reals, which is almost all of the reals.

> sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?

To the extent to which you can form that indexed collection of sets in the first place, surely you can form a similarly indexed collection of elements of them the same way. How can you say you've formed a non-empty set if you can't select an element of it? If we permit ourselves to form this indexed collection "lazily", surely we can do the choice "lazily" as well. (Just my intuition about these things)

Re: The Point of the Banach-Tarski Theorem

#40

Post by podiki »

After a semester of a senior math seminar with one student always bringing up "but you assumed the axiom of choice!" (mostly non-seriously), the gradstudent teaching it decided for his, and last, presentation to show this theorem. It was pretty wild as I had never heard of it before (being a physics-math person). Very cool.

(The seminar was on knot theory but I had more fun with Banach-Tarski).

Post reply on HN