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July 18th, 2016, 05:22 AM   #1
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Banach Tarski Paradox

Banach Tarski paradox is not a paradox because points don't occupy space- they have no size. It takes 2 points to determine a size (measure).

Because points don't occupy space (have no size), I can map the points of any ball into the points of any number of balls, of any size, just like I can map the points of a line into any number of lines of any length.

A point has no dimension.

Thanks to mathman who started this topic:
Reals are uncountable - without Cantor diagonal argument

I took the liberty of starting a new thread because, frankly, I didn't want to see the Banach Tarski Paradox buried.

Last edited by zylo; July 18th, 2016 at 05:28 AM. Reason: changed same size to any size.
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July 18th, 2016, 08:26 AM   #2
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You don't understand anything about Banarch-Tarski or measure theory. Your opinions on the subject are worthless. You have nothing meaningful with which to back up your outdated intuition.
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July 18th, 2016, 11:23 AM   #3
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Divide line 1 into n parts.
Divide line 2 into n parts.

Map p/n on line 1 to p/n on line 2. The mapping holds for any n and in the limit as n -> infinity (countable infinity) all the points on line 1 are mapped to all the points on line 2.
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July 18th, 2016, 11:41 AM   #4
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Quote:
Originally Posted by zylo View Post
Divide line 1 into n parts.
Divide line 2 into n parts.

Map p/n on line 1 to p/n on line 2. The mapping holds for any n and in the limit as n -> infinity (countable infinity) all the points on line 1 are mapped to all the points on line 2.
This has nothing to do with the "Banach-Tarski paradox".
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July 18th, 2016, 12:07 PM   #5
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This has nothing to do with the "Banach-Tarski paradox".
"Doubling the ball" is like mapping points of a line to a line twice it's length.
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July 18th, 2016, 12:50 PM   #6
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Quote:
Originally Posted by zylo View Post
"Doubling the ball" is like mapping points of a line to a line twice it's length.
It's much more interesting than that.

You are correct that the cardinality of a unit ball is the same as that of two unit balls. That's easy to show, it's just an uncountable version of Galileo's paradox that the positive integers are in one-to-one correspondence with the perfect squares.

The Banach-Tarski paradox states that you can decompose the unit ball into as few as five pieces; move those pieces around via rigid motions; and end up with two spheres each having the same volume as the original sphere.

A rigid motion is just what it says. In three-space it's a translation or a rotation, a motion that intuitively preserves volume. That's what makes the paradox interesting.
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