3 ms·

So, Banach-Tarski says you can split a sphere of volume S into a finite number of pieces and reassemble the pieces into two spheres of volume S. Is the proof c

So, Banach-Tarski says you can split a sphere of volume S into a finite number of pieces and reassemble the pieces into two spheres of volume S. Is the proof constructive? As in: does the proof actually show how to build the pieces? If yes, is the boundary of the pieces of measurable surface? Can the pieces be rendered in 3D? Or is it just another one of those "proofs" where if the set of pieces doesn't exist we land on a contraction?

josephcsible 4y ago
https://youtu.be/s86-Z-CbaHA?t=673 https://youtu.be/s86-Z-CbaHA?t=673 nicely visualizes it. It's an infinite number of infinitely thin filaments from the center of the sphere going outward in every direction.
- hgsgm 4y ago
  Not just infinite, uncountably infinite!
ColinWright 4y ago
* The proof uses the Axiom of Choice, so no, it's not constructive; * The boundary of the pieces has to be unmeasurable, in that sense it's similar to Vitali's set[0]; * No, the pieces are more-or-less a "fog" of points. * I don't know why you have "proofs" in quotation marks, and I don't know what you mean by the second half of the sentence (even if I replace "contraction" with "contradiction"). I do feel like a careful reading, or perhaps re-reading, would let you ask more specific questions that we can help you with. [0] https://en.wikipedia.org/wiki/Vitali_set https://en.wikipedia.org/wiki/Vitali_set
- ur-whale 4y ago
  > I don't know why you have "proofs" in quotation marks Because proof by contradiction is viewed by an entire category of mathematicians as "wrong" (please make sure to note the use of quotes in case you missed them). Read all about it here: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_... In a very tangible sense, they're correct. "Proof" by contradiction, tertium non datur, proof by the absurd, whatever you want to call it, more often than not entirely fails to produce working examples, which makes the "proof" fare less useful than the one that explains how to build an actual exemplar.
Smaug123 4y ago
Moreover, there cannot be a constructive proof using this construction, because the Banach-Tarski construction will fail if all sets are measurable (which can happen in the absence of Choice). A constructive proof would continue to succeed in such settings.

josephcsible 4y ago

https://youtu.be/s86-Z-CbaHA?t=673 https://youtu.be/s86-Z-CbaHA?t=673 nicely visualizes it. It's an infinite number of infinitely thin filaments from the center of the sphere going outward in every direction.

hgsgm 4y ago
Not just infinite, uncountably infinite!

ColinWright 4y ago

* The proof uses the Axiom of Choice, so no, it's not constructive; * The boundary of the pieces has to be unmeasurable, in that sense it's similar to Vitali's set[0]; * No, the pieces are more-or-less a "fog" of points. * I don't know why you have "proofs" in quotation marks, and I don't know what you mean by the second half of the sentence (even if I replace "contraction" with "contradiction"). I do feel like a careful reading, or perhaps re-reading, would let you ask more specific questions that we can help you with. [0] https://en.wikipedia.org/wiki/Vitali_set https://en.wikipedia.org/wiki/Vitali_set

ur-whale 4y ago
> I don't know why you have "proofs" in quotation marks Because proof by contradiction is viewed by an entire category of mathematicians as "wrong" (please make sure to note the use of quotes in case you missed them). Read all about it here: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_... In a very tangible sense, they're correct. "Proof" by contradiction, tertium non datur, proof by the absurd, whatever you want to call it, more often than not entirely fails to produce working examples, which makes the "proof" fare less useful than the one that explains how to build an actual exemplar.

Smaug123 4y ago

Moreover, there cannot be a constructive proof using this construction, because the Banach-Tarski construction will fail if all sets are measurable (which can happen in the absence of Choice). A constructive proof would continue to succeed in such settings.