[FOM] Banach Tarski Paradox/Line

George McNulty mcnulty at mailbox.sc.edu
Mon Nov 28 12:05:27 EST 2011

The only thing I might add is that Tarski was amused by the contrast: 
Banach's proof that there are no paradoxical decomposition in dimensions 
one and two relied on the Hahn-Banach Theorem, which in turn relied on 
the Axiom of Choice.  So the result of no paradoxical decompositions was 
a fragile a result as the result giving paradoxical decompostion---from 
the point of view of those who doubted the Axiom of Choice.

Somewhere (in Wagon's book?) I heard there was a proof of no paradoxical 
decompositions in dimensions 1 and 2  without choice, but I do not 
recall where.

George McNulty

On 11/27/2011 04:27 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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