Talk:Banach–Tarski paradox

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Contents 1 Clarifications requested 2 end of proof 3 Curiosity 4 BTP & arguments against axiom of choice 5 Hausdorff paradox 6 Picture? 7 A simple picture 8 Similar Result 9 The Paradox is not related to the axiom of choice 10 endashes 11 Layman's section 12 On hyphenated hyper-2-tuples 13 Doing the decomposition physically

Can this construction (perhaps the one using five pieces) be shown in a picture? Can it be animated on a computer screen (in simulated 3D)?

I find pictures much more intuitive than symbolic manipulation. David 16:10 Sep 17, 2002 (UTC)

All Banach-Tarski constructions involve non-measurable pieces, so I don't think a useful picture (or animation) would be possible. --Zundark 16:46 Sep 17, 2002 (UTC)

What about the series of pictures in Scientific American magazine some years ago that showed how a ball can be sliced up and rearranged to become something else? (I forget what it was.) I believe that example involved non-measurable pieces as well. I think non-measurable pieces can be visualized, just not realized in nature because they may be infinitely thin or whatever. In any case, the pictures were interesting. David 17:05 Sep 17, 2002 (UTC)

A piece which is infinitely thin has Lebesgue measure 0. Non-measurable pieces are much worse than this, and cannot even be explicitly described. I still maintain that a useful picture of a Banach-Tarski dissection is not possible, especially as it's impossible to even specify such a dissection (rather than merely prove one exists). I can't comment on the Scientific American pictures, as I haven't seen them. --Zundark 18:27 Sep 17, 2002 (UTC)

The paradoxical decomposition of the free group in two generators, which underlies the proof, could maybe be visualized by depicting its (infinite) Cayley graph and showing how it consists of four pieces that look just like the whole graph. AxelBoldt 18:41 Sep 17, 2002 (UTC)

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The following text has been copied from Talk:Banach Tarski Paradoxical Decomposition:

How about a full name for Hausdorff so it can be linked?

Is this "doubling the interval" thingie related to the fact that on the Real line there are the same number of points in any interval of any length? Or am I simply showing my ignorance? Seems we need an article on infinity. --Buz Cory

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I've put in Felix Hausdorff's first name, but there's no article for him yet.

Doubling the unit interval would be impossible if the doubled interval didn't have the same number of points as the original. But there's more to it than that, because only countably many pieces are used, whereas breaking it up into individual points would involve uncountably many pieces.

Zundark, 2001-08-09

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[edit] Clarifications requested

Can you please clarify (in the article) the following basic points:

• First, the article should explain what it means by saying that the two balls are of the "same size" as the original, since it implies later in the article that "size" is not meant in the "usual" sense of measure theory.
• Second, it should clearly distinguish the theorem from the "ordinary" fact that any infinite set (e.g. two spheres) can always be identified bijectively with a similarly infinite subset (e.g. one sphere) of itself.
• Third, there seems to be in a contradiction: in the introduction, it talks about a "solid ball" (which sounds like a sphere-bounded volume) whereas the proof talks about S2 (which usually denotes a spherical surface).

Steven G. Johnson 04:07, 22 Mar 2004 (UTC)

• It means size as in Lebesgue measure. Which part of the article do you think implies something different?
  • It says something about the pieces not being measurable, so combined with the fact that it never defines "size" I found this confusing. The main point is that it should clearly define "size". Steven G. Johnson
• It says at the beginning that the decomposition is into finitely many pieces. This clearly distinguishes it from a decomposition into uncountably many pieces. If you don't think this is clear enough, perhaps you could add a clarification yourself.
  • I don't think the article as it stands is clear enough, simply because people might be confused into thinking that the mere identification of two spheres with one is the paradoxical part (most people aren't familiar with the fact that infinite sets can be identified with subsets of themselves). Steven G. Johnson
• The proof talks about S², but at the very end it shows how to extend this to the ball (with a slight fudge). This is also explained at the beginning of the proof sketch, so I don't think it requires further clarification.
  • Sorry, I didn't notice the last sentence. My bad. Steven G. Johnson

--Zundark 10:13, 22 Mar 2004 (UTC)

I've tried to clarify the above points; please check. I'm still a bit confused by your saying that "size" is meant in the sense of Lebesgue measure, since later in the article it talks about there being no non-trivial "measure" for arbitrary sets. I guess the point is that the pieces are not measurable, but their combination is? Steven G. Johnson 21:53, 22 Mar 2004 (UTC)

The revised definition is much more clear, thanks! It is great to have a formal definition of what is actually being proved. Steven G. Johnson 03:12, 23 Mar 2004 (UTC)

Thanks, I think we should not push too hard on these clarifications, it is very nice article, I belive further clarifications might make too havy. Tosha 04:58, 23 Mar 2004 (UTC)

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As I understand the definitions of S(a) in the section about paradox decomposition, we mean . Now note that there exist strings in F₂ which start with "aa..." which means that by aS(a ^{− 1}) we would get the whole free group back and not only the part without S(a ^{− 1}) as indicated by the picture. To point this out: --74.236.150.135

If the first letter is a ^{− 1}, then the second letter can't be a (because we're talking about words in reduced form). So aS(a ^{− 1}) doesn't contain any words starting with a. Note that the picture shows aS(a ^{− 1}) as the complement of S(a), not the complement of S(a ^{− 1}). --Zundark 10:38, 27 November 2006 (UTC)

[edit] end of proof

I removed the last part from the proof, it is a sketch, not a prrof and I think such details should not be covered. Tosha 01:49, 5 May 2004 (UTC)

By the way, there is a new page Hausdorff paradox. I don't feel qualified to work on the content; but it is clearly very close to this page. If this is becoming a featured article candidate, perhaps including tha material might make this page more complete.

Charles Matthews 07:57, 5 May 2004 (UTC)

[edit] Curiosity

I came across an interesting anagram of "BANACH TARSKI".

It's "BANACH TARSKI BANACH TARSKI". —Ashley Y 10:45, 2004 Jul 9 (UTC)

[edit] BTP & arguments against axiom of choice

The usual argument against the idea that the BTP genuinely undermines the plausibility of the axiom of choice, is that the axiom of choice allows one to construct non-measurable subsets, which it is wrong to regard as "pieces" of the original ball: instead they interleave with each other to an infinitesimally fine degree, allowing a trick rather similar to Russell's Hotel to be carried out. What's puzzling is the intrusion of a paradox of the infinite into what at first glance appeared to be a statement of geometry.

I'm not applying an edit directly, because this issue has ramifications elsewhere, and I haven';t time to properly formulate the text right now. Changes are needed:

• cut it up into finitely many pieces -- very misleading description
• it is a paradox only in the sense of being counter-intuitive. Because its proof prominently uses the axiom of choice, this counter-intuitive conclusion has been presented as an argument against adoption of that axiom. -- needs counterargument here
• Changes needed in AofC page ---- Charles Stewart 20:07, 1 Sep 2004 (UTC)

I agree, but it was better, it is all result of the second edit of ArnoldReinhold, so I have reverted it and added all later changes. Tosha 04:06, 2 Sep 2004 (UTC)

I think I've maybe not made my complaint clear: the article begins with an description of the TBP that is couched in intuitive but contentious terms: people who say the resolution of the paradox is that our intuitions about cutting up solids and applying spatial transformations only applies to measurable connected subsets (as indeed I do) will object to the way the topic is framed; the reversion hasn't changed much. I'm not going to have much time for wikiing in the next ten days, but I plan on applying some changes then. ---- Charles Stewart 09:07, 2 Sep 2004 (UTC)

[edit] Hausdorff paradox

The proof here is closer to Hausdorff paradox I think to move it there and leave this page with no proof. Tosha

[edit] Picture?

I can provide an illustration of Step 1 of the proof sketch, namely the paradoxical decomposition of F₂. I envision something like the picture at free group, but with the sets S(a ^{− 1}), aS(a ^{− 1}) and S(a) marked and labelled. Is there interest? --Dbenbenn 01:35, 6 Dec 2004 (UTC)

I think it wold be great (with colors?)... Tosha 02:50, 6 Dec 2004 (UTC)

Done! Is it clear, or can it be improved? --Dbenbenn 04:56, 13 Dec 2004 (UTC)

Very nice I think Tosha 07:18, 13 Dec 2004 (UTC)

[edit] A simple picture

Anyone want this picture on the page?

• Looks good. I'll put it in. Eric119 23:34, 5 Feb 2005 (UTC)

The fact that the free group can be so decomposed follows from the fact that it is non-amenable. I think we should put this in - It makes the discussion a little more transparent.

[edit] Similar Result

I remember reading somewhere about a similar result, the name of which I can't remember. I don't know if the two are related theoretically, but they remind me of each other. The other result says that a set of points exists in euclidean space such that the projections of the set onto different planes can produce any set of 2 dimensional images you want. Does anyone know the name of this, or has anyone even heard of it at all? It seems like a link from this article to one on this other result might be useful. --Monguin61 10:00, 10 December 2005 (UTC)

If anyone else is interested, the idea I was remembering was the digital sundial. Wikipedia's entry on sundials has an external link to the patent of an actual digital sundial, and as of now they are available for purchase from www.digitalsundial.com. The design of the physical device was the result of the work of one K. Falconer, and the idea is described, presumably in more detail, in his book on fractal geometry. --Monguin61 01:23, 15 December 2005 (UTC)

[edit] The Paradox is not related to the axiom of choice

Banach–Tarski paradox is indeed counter intuitive but I'm not sure that what makes it counter intuitive is related to the axiom of choice. I would say that by using the axiom of choice one can get a counter intuitive result from another one which is already counter intuitive :

Think about a subset of the ball as something "covering" some part of the ball. The intuition tells us that when you move by a rotation a subset of the ball into the ball you will "uncover" some points and also "cover" some points which where "uncovered" before applying the rotation, but that these two phenomena will compensate each other. In fact this is not true in the non measurable world and this is the heart of the Banach–Tarski paradox.

More precisely the intuition tells us that it is impossible to find a subset X of the unit sphere (the sphere bounding the ball), apply a rotation to it and get a subset strictly included in X.

Similarly intuition tells us that it is impossible to find a subset X of the unit sphere which is strictly included in the sphere, apply a rotation to X and get something which stricly contains X.

This is nevertheless true :

Using the same notation as in the article take some x lying in the sphere (not on the rotation axis) and consider the subset of the sphere X = S(a ^{− 1}).x. If one applies the rotation a ^{− 1} to it we get something stricly included in X and if one apply the rotation a to X one get something that strictly contain X. This is due to the fact that

and the way H is acting on the unit sphere.

You should realize that the Banach-Tarski paradox is only somewhat more strange than the fact that e.g. applying a shift to the left by 10 units of length to the set of all integers greater than 10 on the real line (and labeling corresponding points with their new values) produces a strictly bigger set - the set of all positive integers.

Hi, new to wikipedia so not totally sure how to talk in here, apologies. Just to pick up on this; the set of all positive integers is not a bigger set than the set of all integers greater than 10. Intuitively it seems to be, but there is a bijection between the two sets given by the function, i.e. f(x) = x - 10. So you can pair off one number from each set together: (1,11), (2,12), (3,13),... etc. In this way, the two sets have the same size. So the Banach-Tarski result is actually much stronger. Bobmonkey 19:27, 13 November 2006 (UTC)

[edit] endashes

Someone recently moved this article from Banach–Tarski paradox to Banach-Tarski paradox (substituting the endash with a hyphen). Really that's the name I'd prefer too; I don't like all these Unicodes that look almost like ASCII characters and can easily be mistyped. But the current WP standard seems to be endashes when the names of two workers are combined, so I moved it back. --Trovatore 23:57, 4 January 2006 (UTC)

I moved it back to a hyphen. I don't know what "current WP standard" you're referring to, but I've never seen an en dash used in this type of case anywhere else. In particular, in TeX it's terribly easy to make an en dash (just type --), but even in TeX people use just a hyphen. dbenbenn | talk 12:01, 15 January 2006 (UTC)

Well, the case has been made that endashes are the correct way, so that we can know that Banach and Tarski are two people. I have come round to agreeing, despite the potential inconveniences. Birch–Swinnerton-Dyer then uniquely parses to two people. So, please don't move back like that. No one should get fanatical about format issues, but making a stand on them is just going to become a time-sink and distraction. Charles Matthews 19:30, 15 January 2006 (UTC)

The question is, which use is more common? It isn't up to us to determine whether an en dash is "correct". And in my experience, Banach-Tarski paradox is always written with a hyphen. I have never seen it written with an en dash. (Whereas mathematical papers written in TeX always use an en dash in something like Birch–Swinnerton-Dyer conjecture, precisely because using a hyphen would be ambiguous.) dbenbenn | talk 09:37, 16 January 2006 (UTC)

I have moved the page back to a hyphen for the aforestated reasons. What is more intuitive to type? wiki/Banach-Tarski paradox or wiki/Banach%E2%80%93Tarski_paradox?Kyle McInnes (talk) 15:40, 20 October 2006 (UTC)

It's irrelevant what's easier to type, as long as there's a redirect. Why do people keep bringing up the typing thing? That's what redirects are for. --Trovatore 17:28, 20 October 2006 (UTC)

Apologies. Thanks for creating the redirect. Kyle McInnes (talk) 17:41, 20 October 2006 (UTC)

No prob. For future reference, redirects are automatically created by page moves; it was there before this latest move-and-move-back. --Trovatore 18:59, 20 October 2006 (UTC)

The claim is that the endash is standard, in text prepared going through a professional copy-editing process. But, as I say, I don't think time should be spent on warring over such issues. Charles Matthews 10:14, 16 January 2006 (UTC)

As far as I understand, it should be with endash. (One can do anything in his TeX-file but for encilopedia should follow standards, also responsible editor would change it for publication). Tosha 16:17, 19 January 2006 (UTC)

[edit] Layman's section

This is a popular paradox, and I believe that there are (relatively) many laypersons who may have a passing interest in what this thing is all about. In my opinion, it would be nice if there was a section here that would require very little background, intuitively explaining what the thing is about and wouldn't require much time to read (no equations!).

Sketch of how such a section could be done:

• Size comparison. Explain that if there is a one-to-one correspondence (a bijective function) between two sets, they aren't necessarily of the same size in the ordinary sense. Example about [0, 1] <--y=2x--> [0,2]. This is why measure theory is necessary. Measure theory gives a notion of size (measure) that works much like Riemann integrals do (I assume the layperson knows some basics about integrals, if he doesn't, all explanation is futile), but is more general. Even so, the "problem" with measure theory is that not all sets are measurable, just like not all functions can be integrated. These non-measurable sets are weird beasts for which our intuition does not work. (Are all/most fractals measurable? Would be good example either way, as they're also famous "weird geometry" beasts) See also Cantor set for a set with infinite number of points but measure of 0 (infinite amount of "Cantor dust" is still just dust; similarly, a curve has measure of 0 in two dimensions, because it does not have area, except for weird beasts).
• Show that a set of points can easily be duplicated, if there are no restrictions as to what you can do with it. E.g. take a unit cube, cut it in two at x=0.5, then (translate and) multiply x of each point in the set with two, giving two exactly same sets of points as the original cube. This works the same way as one-to-one correspondence way of comparing set size: you might think that scaling x this way would leave "gaps" in the end result, but this is not so. Infinite sets are weird enough as they are. But the Banach-Tarski paradox is more paradoxical than just this.
• The paradox in Banach-Tarski paradox is that if you take a (solid) sphere, cut it in 5 pieces (although non-measurable, weird beast pieces), then move and rotate these to their places, you get two exact copies of the original sphere. Being exact copies, they of course have the same measure, too. Note that there is no scaling here, and no carving of the curve to infinite number of pieces, or anything. All the "funny stuff" is in what the 5 pieces look like. Being non-measurable, such pieces of course can't exist in the real world, as all real world objects are measurable (we can't even carve things up to infinite precision, much less somehow invoke the axiom of choice).
• Of the five pieces, one is for "fixing up" a small (countable) amount of gaps in the actual construction done with the other four pieces. The idea is to split the sphere in 4 parts in such a way that you can reassemble the original sphere from two of them, thus getting two original spheres. This reassembly is possible, because two of the pieces, which are supposed to be one-quarter of the size of the original sphere, actually become 3/4ths of the sphere when rotated by about 70.5 degrees. By performing this rotation on both of them, we get two 3/4ths of the sphere, and still have two 1/4 pieces to match. Combining these, we get two full spheres. However, this is merely an intuitive description, as the pieces are in fact non-measurable, and thus a notion that they are "quarter" or "3/4th" of the original sphere has no meaning. The construct thus escapes from the common fact that rotation preserves size by rotating pieces that do not have size. It just so happens, that when these pieces are combined to again produce measurable objects, the total size has magically doubled in the process.
• For details, see the actual proof.

I don't know how exactly to phrase this, and my knowledge of the matter is rather superficial (I'm pretty much a layperson myself!), so I didn't dare actually try to modify the article.

The goals I was thinking of are:

• Do not require much knowledge. (high school math is enough)
• Provide (non-obscure!) references for further reading, to connect all this to a larger background.
• Be clear in what is paradoxical and counterintuitive. This means explicitly pointing such things out.
• Be clear in what is not paradoxical or counterintuitive. This means explicitly stating what part the paradox is restricted to. "At least I don't have to think about these parts" makes for easier comprehension.
• Be clear in what the Banach-Tarski paradox is about, as opposed to what other (semi-)counterintuitive results there are.
• Try to keep reader's interest instead of merely stating the facts.

Too much to ask? Maybe. But in my opinion, an explanation like stated above would exist in an ideal encyclopedia entry, as not all interested readers know much of anything about modern math. (and an ideal encyclopedia is for everyone, right? :-)

It may be that some of the above would be better to put in a general article about geometrical paradoxes (and a link from here to that), but I think that merely referring to the articles about measure theory, related paradoxes, etc. would scare many away. Reading a layperson's section should require just about the same amount of effort as reading a popular science magazine article, and chasing hyperlinks in search of comprehensible and relevant information is a far cry from that.

130.233.22.111 16:35, 6 February 2006 (UTC)

[edit] On hyphenated hyper-2-tuples

• JA: You know I really hate being so WikiPiki, but there's supposedly a good reason to use ndashes instead of hyphens in names that refer to multiple persons rather than multiple ancestors of one person, so Banach-Tarski paradox should be Banach–Tarski paradox, and Hausdorff-Banach-Tarski paradox should be Hausdorff–Banach–Tarski paradox, and so on. I would move the article ¼-with myself, but there's already a redirect from Banach–Tarski paradox to Banach-Tarski paradox, so it takes an admin-assisted deletion of Banach-Tarski paradox to do that. Are you beginning to understand how the same content can get itself redistributed across two identical contents now? Jon Awbrey 16:02, 9 March 2006 (UTC)

I went ahead and moved it (no admin assistance required). You can't move an article onto another real article, but you can move article A onto article B when B is a redirect to A, unless B has some other history. Not sure what the rule is if B has at some point been a redirect to C. --Trovatore 17:13, 9 March 2006 (UTC)

[edit] Doing the decomposition physically

In a recient article in the Journal of Symbolic Logic, Trevor M. Wilson titled "A continuous movement version of the Banach—Tarski paradox: A solution to de Groot's Problem", it was proved that the dissaembly and reassembly of the balls can be performed by continuously and rigidly moving the pieces without the pieces ever intersecting at any point in time. Should this fact be mentioned and referenced? --Ramsey2006 22:46, 13 October 2006 (UTC)

The pieces themselves are still infinitely convoluted and cannot actually be constructed physically (atoms are of a finite size, after all). So this still doesn't get anywhere near doing the decomposition "physically". But the fact that they can be translated, rotated, and reassembled without ever intersecting may be worth the mention.—Tetracube 06:01, 15 October 2006 (UTC)

Done. I also removed a confused paragraph about it not being a real paradox, which in addition to being dubious was also in a completely inappropriate section. --Trovatore 06:40, 15 October 2006 (UTC)