July 18th, 2016, 06:22 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,214 Thanks: 91  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 06:28 AM. Reason: changed same size to any size. 
July 18th, 2016, 09:26 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,116 Thanks: 2369 Math Focus: Mainly analysis and algebra 
You don't understand anything about BanarchTarski or measure theory. Your opinions on the subject are worthless. You have nothing meaningful with which to back up your outdated intuition.

July 18th, 2016, 12:23 PM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,214 Thanks: 91 
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. 
July 18th, 2016, 12:41 PM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,919 Thanks: 785  This has nothing to do with the "BanachTarski paradox".

July 18th, 2016, 01:07 PM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,214 Thanks: 91  
July 18th, 2016, 01:50 PM  #6  
Senior Member Joined: Aug 2012 Posts: 1,678 Thanks: 436  Quote:
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 onetoone correspondence with the perfect squares. The BanachTarski 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 threespace it's a translation or a rotation, a motion that intuitively preserves volume. That's what makes the paradox interesting.  

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