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

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.

"Pawlikowski proved..."

Article sez:

In 1991, using then-recent results by Matthew Foreman and Friedrich Wehrung, Janusz Pawlikowski proved that the Banach–Tarski paradox follows from ZF plus the Hahn–Banach theorem. The Hahn–Banach theorem does not rely on the full axiom of choice but can be proved using a weaker version of AC called the ultrafilter lemma. So Pawlikowski proved that the set theory needed to prove the Banach–Tarski paradox, while stronger than ZF, is weaker than full ZFC. [emphasis mine]

The bolded claim appears to be technically true, but it suggests that it wasn't known before 1991 that full ZFC was not needed to prove Banach–Tarski, which is certainly false even leaving aside the quibble that ZFC is not finitely axiomatizable so its full strength is never needed to prove any particular ZFC theorem. It would have been obvious from the beginning that a wellordering of R is enough choice to prove Banach–Tarski, and it must have been known from the early days of forcing that a wellordering of R does not imply full AC. --Trovatore (talk) 22:58, 28 January 2024 (UTC)

Update: I happened to glance up the page and notice that I had made essentially the identical complaint sixteen years ago. I hope I've had at least one original thought in that time. Anyway I guess it's time to stop soliciting opinions and do something about it. I just removed the last sentence of the quoted paragraph. --Trovatore (talk) 04:49, 30 January 2024 (UTC)

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)