[FOM] Banach Tarski Paradox/Line

[K. P. Hart]

On Sun, Nov 27, 2011 at 10:27:30PM +0100, 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?

Banach proved that Lebesgue measure on the real line has a finitely additive
extension to the family of all subsets.
This extension is also invariant under isometries.

http://matwbn.icm.edu.pl/ksiazki/fm/fm4/fm412.pdf

This makes a finite decomposition impossible.

KP Hart


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