[FOM] Re: countable choice and the Banach-Tarski paradox

[Ali Enayat]

This is a belated reply to H. Nordmark's query [Sep 5, 2004]:

"Can the Banach-Tarski paradox be derived from the Axiom of Countable
Choice?".

1. The answer is no, at least assuming the consistency of (ZFC + there is an
inaccessible cardinal). This is a consequence of the fundamental work of
Solovay [1970, Annals of Math], who showed that

Con(ZFC + there is an inaccessible) implies Con(ZF + DC + all sets of reals
are Lebesgue measurable).

Note that (A) the "paradoxical" sets in the Banach-Tarski theorem have to be
non-measurable, and (B) DC [dependent choice] implies the axiom of countable
choice.

2. This leads to the question I would like to pose:

Does Con(ZF) imply Con(ZF + DC + "there is no paradoxical Banach-Tarski
decomposition of the unit ball").

3. Note that Shelah's model [Israel Journal of Math, 1984], constructed only
from Con(ZF), in which DC holds and all sets of reals have the Baire
property, is of no help in answering the above question since Dougherty and
Foreman [ J. Amer. Math. Soc., 1994] have shown that there are paradoxical
decompositions of the unit ball using pieces which *have the property of
Baire*. I do not know how much choice is needed in their construction.

Regards,

Ali Enayat