[FOM] Banach Tarski Paradox/Line

David Roberts david.roberts at adelaide.edu.au
Sun Nov 27 23:39:53 EST 2011

The Banach-Tarski 'paradox' arises because SO(3) x |R^3 < Aut(S^2)
contains a free group on 2 letters
(or more accurately, it is not amenable) and similarly for SO(n) for all n > 2.
To arrive at the version for B^3 I believe one can take a
decomposition of the ball into concentric spheres
and somehow take a limit, but this shouldn't be necessary.

Terry Tao has a nice blog post on this:


in which he makes the distinction between 'finitely equidecomposable'
and 'infinitely equidecomposable'.

The BT paradox falls into the former case, and it turns out that [0,1]
is the latter but _not_ the former.
Note that one must also specify the group of automorphisms under which
one wants to
'move around' the pieces.

David Roberts

On 28 November 2011 07:57,  <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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