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

Here Con($X$) is the consistency of $X$; $DC$ is dependent choice.

Motivation for the Question: Since the "paradoxical" sets in the Banach-Tarski theorem have to be non-measurable, Solovay's model of "every subset of reals is measurable" shows:

Theorem. Con($ZF$+ Inacc) implies Con($ZF$ + $DC$ + "there is no paradoxical Banach-Tarski decomposition of the unit ball").

In the above Inacc is the statement "there is an inaccessible cardinal".

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.

This question was posed a while ago on an FOM-posting of mine, but remained unanswered.

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Ali, are you implicitly taking a position on the consistency of inaccessible cardinals? After all, if we believe Con(ZF+Inacc) is true, then the material implication of your question would follow by the Solovay result. But probably you mean to ask whether the implication is provable in some specific weak theory, such as ZFC or much weaker, rather than whether it is true. – Joel David Hamkins Aug 12 '11 at 17:06
Joel: My question is motivated on the question whether "there is no paradoxical decomposition" has large cardinal strength. All relative consistency results are verifiable in Primitive Recursive Arithmetic (including the forcing ones), but for the purposes of my question, we can take the meta-theory to be PA or even second order arithmetic. – Ali Enayat Aug 12 '11 at 17:36
Yes, Ali, I was merely trying to tease you, since you appeared to ask the question as a material implication. – Joel David Hamkins Aug 12 '11 at 18:07
Joel: I see, I guess I was being too literal in answering you; but others may find this literal answer useful in clarifying the question. – Ali Enayat Aug 12 '11 at 18:32
Although I haven't read the Dougherty--Foreman paper, I remember Kechris telling me that their construction was effective, so this confirms your suspicion that Shelah's model does not answer your question. – Trevor Wilson Jul 17 '12 at 20:08
up vote 8 down vote accepted

A positive answer is proved in S. Wagon's book "The Banach-Tarski Paradox", Theorem 13.2. Specifically, the statement proved there is:

Con(ZF) $\leftrightarrow$ Con(ZF + DC + GM),

where GM is the existence of an isometry-invariant measure on all subsets of $\mathbb R^n$ taking the value $1$ on the unit cube.

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According to Wagon, the model of ZF+DC+GM is obtained the same way as Solovay's except only after adding $\aleph_1$ random reals. – François G. Dorais Oct 15 '13 at 2:18

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