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 April 6th, 2017, 11:53 AM #1 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 Banach-Tarski Paradox A flying cow is not a paradox, it is a figment of the imagination, "If I can imagine it, it exists" notwithstanding. An infinite number of points on a line is a figment of the imagination. Real points have size. So unravelling the Banach-Tarski "paradox" is quite easy: Start with a figure made up of points. Stop right there. You can't do it. Points don't have size, they don't exist. But you can imagine they exist and your results will be imaginary. There is nothing mysterious or magical or paradoxical about it. Geometry deals with a line from a to b. Not a line consisting of n points. There are not n points in a circle. You cannot create a line or circle out of points. Classical analysis doesn't deal with numbers of points, it deals with increments. $\displaystyle \Delta$x is not defined by a number of points.
April 6th, 2017, 02:02 PM   #2
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Quote:
 Originally Posted by zylo A flying cow is not a paradox, it is a figment of the imagination, "If I can imagine it, it exists" notwithstanding. An infinite number of points on a line is a figment of the imagination. Real points have size. So unravelling the Banach-Tarski "paradox" is quite easy: Start with a figure made up of points. Stop right there. You can't do it. Points don't have size, they don't exist. But you can imagine they exist and your results will be imaginary. There is nothing mysterious or magical or paradoxical about it. Geometry deals with a line from a to b. Not a line consisting of n points. There are not n points in a circle. You cannot create a line or circle out of points. Classical analysis doesn't deal with numbers of points, it deals with increments. $\displaystyle \Delta$x is not defined by a number of points.
In terms of Physics you are absolutely right. In terms of Math you are essentially going all the way back to the sources that Euclid used to write up The Elements. Have you taken a gander there to see if Euclid defined a line to be made up of a finite number of elements?

-Dan

 April 6th, 2017, 06:24 PM #3 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 But the premise and conclusion are expressed in physical terms. From wiki: "A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the sun." If the objects are considered to consist of an infinite number of points, that's trivial.
April 6th, 2017, 06:56 PM   #4
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Quote:
 Originally Posted by zylo But the premise and conclusion are expressed in physical terms. From wiki: "A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the sun." If the objects are considered to consist of an infinite number of points, that's trivial.
The pea and the sun, which is actually the name of a book, is terribly misleading. I oppose anyone who makes this kind of sloppy metaphor that confuses people about what the Banach-Tarski theorem is actually about, which is abstract mathematical 3-space only. The author of that book is wrong and the Wiki editor who quoted the title (without attribution) is wrong.

It's like the rubber sheet and bowling ball visualization of relativistic gravity. It's a popularization, not to be taken literally. As far as we know it's not true about the world. Of course if tomorrow morning professor so and so announces the discovery of a copy of the free group on two letters in the physical world ... all bets are off. [That would involve finding an actual infinity in the world].

Last edited by Maschke; April 6th, 2017 at 07:09 PM.

 April 6th, 2017, 07:05 PM #5 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 What is your version of Banach Tarski Paradox? Never mind. Don't waste your time. I assume you accept: http://mathworld.wolfram.com/Banach-TarskiParadox.html Which is just as ridiculous as the sun and the pea without a definition of ball. If the ball is undefined, we are dealing with flying cows. Last edited by zylo; April 6th, 2017 at 07:13 PM.
April 6th, 2017, 07:41 PM   #6
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Quote:
 Originally Posted by zylo What is your version of Banach Tarski Paradox? Never mind. Don't waste your time. I assume you accept: Banach-Tarski Paradox -- from Wolfram MathWorld Which is just as ridiculous as the sun and the pea without a definition of ball. If the ball is undefined, we are dealing with flying cows.
What makes you say that the unit ball in three space is not defined? It's defined perfectly well as $\{x \in \mathbb R^3 : \| x \| \leq 1\}$ where $\| x \| = \sqrt{x_1^2 + x_2^2 +x_3^2}$ is the usual Euclidean distance.

Last edited by Maschke; April 6th, 2017 at 07:45 PM.

 April 6th, 2017, 08:08 PM #7 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 That's not a ball. That's the location of points in a ball. You can't destroy your coordinate system. If you cut up the ball, to what coordinates do the components go. Presumably distances remain unchanged. Or is this what you are talking about? [0,1] H [0,2] small "ball" = large "ball" [0,2] H [0,2]U[2,4]U[4,6] one "ball" = three "balls" .
April 6th, 2017, 08:15 PM   #8
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Quote:
 Originally Posted by zylo That's not a ball. That's the location of points in a ball. You can't destroy your coordinate system. If you cut up the ball, to what coordinates do the components go. Presumably distances remain unchanged. Or is this what you are talking about? [0,1] H [0,2] small "ball" = large "ball" [0,2] H [0,2]U[2,4]U[4,6] one "ball" = three "balls" .

No, B-T uses are isometries, rigid motions. Distance-preserving maps. Your homeomorphisms don't preserve distances. They are cardinality equivalences but not measure equivalences.

 April 6th, 2017, 08:21 PM #9 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 OK. So are we talking rigid motions (about an axis) or "rigid motions." I assume after ball is cut apart, the components are rigidly (distance preserving) moved somewhere in the coordinate system. How many parts, and how many rigid motions? If infinite, all bets are off. Last edited by zylo; April 6th, 2017 at 08:27 PM.
April 6th, 2017, 08:48 PM   #10
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 Originally Posted by zylo So are we talking rigid motions (about an axis) or "rigid motions."
Yes, rigid rotations about an axis. The proof uses two rotations, one about the z-axis and one about the x-axis, and all the combinations of those two rotations.

Quote:
 Originally Posted by zylo I assume after ball is cut apart, the components are rigidly (distance preserving) moved somewhere in the coordinate system.
Yes. Every point in the original ball gets moved somewhere, and all distances are preserved.

Quote:
 Originally Posted by zylo How many parts, and how many rigid motions? If infinite, all bets are off.
It can be done with as few as five pieces; but I believe the Wiki proof I'm working through needs more pieces, but still finite. I'm not sure what it means to ask how many motions. You can break the ball up into 5 pieces, move the pieces around with a rigid motions, and end up with two balls each the size of the original. The proof involves the axiom of choice so there's a nonconstructive aspect to it. There's no algorithm. It's an existence proof. There's no sequence of steps. It's a proof that such a decomposition exists. It's not something we can ever visualize. Nobody can visualize a nonmeasurable set.

Last edited by Maschke; April 6th, 2017 at 09:04 PM.

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