My Math Forum Banach Tarski Paradox

 Topology Topology Math Forum

 July 18th, 2016, 05:22 AM #1 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 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.
 July 18th, 2016, 08:26 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra 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 You have nothing meaningful with which to back up your outdated intuition.
 July 18th, 2016, 11:23 AM #3 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 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, 11:41 AM   #4
Math Team

Joined: Jan 2015
From: Alabama

Posts: 3,160
Thanks: 866

Quote:
 Originally Posted by zylo 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".

July 18th, 2016, 12:07 PM   #5
Senior Member

Joined: Mar 2015
From: New Jersey

Posts: 1,364
Thanks: 100

Quote:
 Originally Posted by Country Boy 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.

July 18th, 2016, 12:50 PM   #6
Senior Member

Joined: Aug 2012

Posts: 1,888
Thanks: 525

Quote:
 Originally Posted by zylo "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.

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post raul21 Applied Math 9 July 18th, 2016 01:25 PM PeterPan Real Analysis 1 June 29th, 2013 07:57 AM Cogline Real Analysis 3 January 28th, 2010 01:53 PM aptx4869 Real Analysis 3 October 25th, 2008 02:13 PM Nevski Real Analysis 1 February 26th, 2008 10:33 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top