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army1987 comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion

10 Post author: XiXiDu 06 January 2012 05:23PM

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Comment author: [deleted] 06 January 2012 11:39:27PM 2 points [-]

This Banach-Tarski explanation is nice at a very beginner level, but worse than useless above that. Here is a very important related fact: The Banach-Traski paradox is simply NOT TRUE on the line and the plane. You can not do such a rearrangement with a circle to get two equally sized circles.

I seem to recall reading about a way to divide the interval [0, 1] in subsets, translating some of them, and getting [0, 2] (involving the Vitali set or something like that), but maybe my memory fails me.

These were the main ideas. One very minor idea is that if you have a paradox for the sphere using rotations, you can get a paradox for the ball. This is a nice homework.

Whfg pbafvqre gur onyy nf orvat znqr hc ol enqvv. Or am I missing something?

Comment author: DanielVarga 07 January 2012 01:32:53PM 2 points [-]

I seem to recall reading about a way to divide the interval [0, 1] in subsets, translating some of them, and getting [0, 2] (involving the Vitali set or something like that), but maybe my memory fails me.

This is possible if you use infinitely many subsets. With an uncountably infinite number of pieces it is true by definition, with a countably infinite number of pieces it can be proven using the Vitali set, and with a finite number of pieces it is not true.

Or am I missing something?

What Oscar_Cunningham said, but basically, no, you are not.

Comment author: [deleted] 07 January 2012 02:10:46PM 0 points [-]

Or am I missing something?

What Oscar_Cunningham said, but basically, no, you are not.

I was expecting something harder given that you called it "a nice exercise", so I pretty much assumed that mine was not the right solution...

Comment author: DanielVarga 08 January 2012 12:55:39AM 0 points [-]

Okay. In the original formulation of the paradox, the task is to cut a ball into pieces, and assemble two balls from the pieces. If I am not mistaken, you have solved a slightly easier task: cut a ball into pieces, and covered two balls with the pieces (with overlaps). A part of the "nice exercise" is to bridge this gap.

Comment author: Oscar_Cunningham 07 January 2012 11:35:37AM 1 point [-]

Jurer qbrf gur prager tb?

Comment author: [deleted] 07 January 2012 12:09:16PM 0 points [-]

Jvgu gur abegu cbyr (be nal bgure neovgenel qverpgvba). Vf gurer nal ceboyrz jvgu gung V pnaabg frr?

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