Question about the Banach–Tarski paradox.

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Discussion Overview

The discussion revolves around the Banach–Tarski paradox, exploring its implications and relationship to dimensionality and measure theory. Participants examine the concept of constructing shapes from points and the nature of non-measurable sets.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether the Banach–Tarski theorem allows for constructing a cube from a square by picking out points, drawing a parallel to the theorem's implications.
  • Another participant clarifies that while a line can be transformed into a square, the Banach–Tarski paradox specifically allows for a sphere to be divided into five pieces and reassembled into two spheres, which does not apply to lower dimensions.
  • Some participants note that it has been proven that such transformations cannot occur in one or two dimensions.
  • A participant proposes viewing the points of a sphere as infinitesimally small cubes and questions the feasibility of constructing multiple spheres from these cubes, challenging the understanding of the theorem's application.
  • Another participant emphasizes the significance of the theorem's reliance on Euclidean transformations and the complications arising from non-measurable sets, suggesting that the behavior of measure is not preserved in such cases.
  • There is a mention of previous pseudo-paradoxes that failed to account for measure-preserving transformations, contrasting them with the Banach-Tarski paradox.
  • One participant expresses a need to further explore the theorem and measure theory.

Areas of Agreement / Disagreement

Participants generally agree on the fundamental aspects of the Banach–Tarski paradox and its implications for higher dimensions, but there are competing views regarding the interpretation of the theorem and its application to constructing shapes from points. The discussion remains unresolved on certain speculative aspects.

Contextual Notes

Participants highlight limitations in understanding the theorem and its reliance on non-measurable sets, as well as the complexities involved in measure theory. There are unresolved questions about the implications of viewing points as infinitesimal cubes.

cragar
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I was reading a little bit about the Banach–Tarski theorem. Is this similar to a line segment of length 1 having the same points as a square with side lengths of 1. And then also a cube with sides of length 1. So then I should be able to take a square and pick out all the points and construct a cube with the same side length. And I should be able to construct as many cubes as I want from that square just by picking out points and constructing my cube. Is this related to the Banach–Tarski theorem or am I crazy.
 
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cragar said:
I was reading a little bit about the Banach–Tarski theorem. Is this similar to a line segment of length 1 having the same points as a square with side lengths of 1. And then also a cube with sides of length 1. So then I should be able to take a square and pick out all the points and construct a cube with the same side length. And I should be able to construct as many cubes as I want from that square just by picking out points and constructing my cube. Is this related to the Banach–Tarski theorem or am I crazy.

Hi cragar! :smile:

The Banach-Tarski paradox is a little bit stronger. When you have a line, you can indeed form a square out of this line. However, what we do there is we take every point separately and map it to a point in the square. So we have to cut our line into an infinite number of pieces and then reassemble it.
Banach-Tarski is a lot stronger: it says we can cut our sphere into \mathbf{5} pieces, and then reassemble it to form two balls. You can't do this with the line: you can't take 5 line pieces and reassemble it to form a square! But you can do it with a ball.
 
so we can do it with a ball but not a line or square.
 
cragar said:
so we can do it with a ball but not a line or square.

Indeed, it has been proven that we can't do this in one or two dimensions.
 
thanks for your answers by the way. Ok let's say I have a sphere of radius 1, can i view the points as infinitesimally small cubes? Then from these cubes I could construct 2 other spheres of radius 1. You said that the theorem cuts the sphere into 5 parts. Why can't I just say when I pick my little cubes from the sphere that I do it 2, 3, or how ever many ways and I put these cubes in a box. So I have an infinite amount of cubes in each box and I might have 3 boxes, then I use these 3 boxes to construct 2 other spheres of radius 1, And the boxes represent my finite area partitions of the original sphere.
 
The two key qualitative facts of the Banach-Tarski paradox are:
  • The motions are simple -- it uses Euclidean translations and rotations (volume-preserving operations) on finitely many objects
  • The sets involved are so "complicated" that the notion of volume doesn't have any meaning for them (they're called non-measurable sets)

(aside: there are lots of "measures" -- notions like "how many", "length", "area", and "volume" are all different sorts of measures)

The first point is rather important -- without it (or something similar), there's no reason to believe that such an argument would preserve measure. As you point out, it's a rather simple matter to take the points of a line and rearrange them into a square -- but the way you do it gives us no reason to think that it should preserve measure*


Previous pseudo-paradoxes that properly use measure-preserving transformations had other factors against them that make it easy for people to mentally brush off the use of non-measurable sets and simply ascribe any poor behavior of measure to the ways in which the argument is complicated.


The Banach-Tarski (pseudo-)paradox is significant because there is pretty much no room to rationalize things away -- it really does a good job of forcing people to acknowledge non-measurable sets and just how badly the idea of measure behaves in their presence.

(Of course, this acknowledgment leads some people to adopt versions of set theory in which non-measurable sets don't exist)


*: well, we have reason to believe the counting measure is preserved, and it is. (+\infty for both a line and for a square)
 
I guess I need to read more about the theorem and measure.
 
cragar said:
I guess I need to read more about the theorem and measure.

Maybe read my blog post about it: https://www.physicsforums.com/blog.php?b=2993
It might help...
 
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