Talk:Banach-Tarski paradox

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)

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

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)

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)

Curiosity

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

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

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)

Hausdorff paradox

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

Picture?

I can provide an illustration of Step 1 of the proof sketch, namely the paradoxical decomposition of <math>F_2<math>. I envision something like the picture at free group, but with the sets <math>S(a^{-1})<math>, <math>aS(a^{-1})<math> and <math>S(a)<math> 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)

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.