FOM: BT and Borel

David Ross ross at
Mon Dec 8 15:54:38 EST 1997

A couple of posters have suggested eliminating the Banach-Tarski
paradox as one advantage of restricting to Borel sets.  I think
that while true, this is misleading - what's nasty about BT is not
so much the paradoxical volume decomposition, but the existence
within the Euclidean group on R^3 of a free group on two generators.
(This is why there's no BT in R^2.)  If we restrict to the Borels this
free group doesn't go away; we merely lose one its more striking (and
probably inoffensive) consequents.

- David (ross at

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