Talk:Banach–Tarski paradox

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 $F_2$. 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)

A simple picture

File:Doubling of a sphere, as per the Banach-Tarski Theorem.png

The Banach-Tarski "paradox": A sphere can be decomposed and reassembled into two spheres the same size as the original.

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.

Von Neumann Paradox section and Recent Progress subsection

I just tweaked some references in the Recent Progress subsection of the Von Neumann Paradox section and I noticed several things:

1. The Von Neumann Paradox section without the Recent Progress subsection is comparable in size to the main article on the Von Neumann Paradox; if the Recent Progress subsection is included it may actually be slightly longer. This seems unbalanced to me. Perhaps much of the material on the Von Neumann Paradox could be moved to the main article on that topic?
2. It seems that most of the Recent Progress items are related to the Von Neumann Paradox, but at least a few—perhaps the last three?—are related to the Banach–Tarski Paradox itself. The latter probably should be moved out of this section.
3. The chronological, timeline structure is not encyclopedic, in my opinion. One of the tweaks I made was to remove a remark that some 2024 work was an extension of some 2017 work by the same author. I removed it because it seemed a minor point and not very informative. I now see that the 2017 work was one of the earlier timeline items—something I missed initially because the reference was duplicated rather than linked. This suggests to me that the timeline structure isn't the right one for this material. A logical presentation by topic would be better as it would make the links between related results easier to talk about.
4. "Recent" tends to fall out of date quickly—another reason for doing away with the timeline structure.
5. It's not clear to me, a non-expert, how important these results are to include in an encyclopedia article. This section is based mostly on primary rather than secondary sources. Secondary sources to establish context would be helpful, if they exist

Will Orrick (talk) 02:37, 19 July 2024 (UTC)

Help me understand why?

"Now let A be the original ball and B be the union of two translated copies of the original ball. Then the proposition means that the original ball A can be divided into a certain number of pieces and then be rotated and translated in such a way that the result is the whole set B, which contains two copies of A" Agreed, but no longer solid, so paradox fails, as objects while resemble A are not same mass etc.121.98.30.202 (talk) 15:25, 13 May 2025 (UTC)

Solidity and mass are properties of matter. This is a geometric property of point sets. If by "no longer solid" you mean the balls contain holes or missing points, that is not the case. Anyway, there are probably better sites to get answers to questions like this than a Wikipedia talk page. Will Orrick (talk) 17:42, 13 May 2025 (UTC)

1. REDIRECT Template:BreakYou could ask on the Wikipedia mathematics reference desk, WP:RD/MATH. --Trovatore (talk) 19:06, 13 May 2025 (UTC)