[FOM] Banach Tarski Paradox/Line

Brian White white at math.stanford.edu
Mon Nov 28 00:19:08 EST 2011

It is not possible in one or in two dimensions.
The obstacle is that the group of isometries
(of R or of R^2) is solvable.

Stan Wagon's book "The Banach-Tarksi Paradox"
covers these and many related matters very nicely.

Sierpinski's old book "Congruence of Sets" is also
wonderful.  In general, Wagon's book goes considerably
farther, but Sierpinski's has a few nice things Wagon
does not cover.

There is a Banach-Tarski-like paradox for one-dimensional intervals
that was discovered by von Neumann.  I beileve
that Wagon and Sierpinski both describe it.


On Nov 27, 2011, at 1:27 PM, pax0 at seznam.cz wrote:

> Is the Banach Tarski paradox provable for the unit real interval; 
> i.e. is there a possibility for duplicating [0,1].
> If not, where is the obstacle?
> Jan Pax
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