TITLE: Is there any version of the Banach-Tarski paradox in ZF?
QUESTION [38 upvotes]: The Banach-Tarski paradox states that for a solid ball in 3‑dimensional space, there exists a decomposition into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original one.
Obviously it is based on AC. I was wondering if anyone here knew if analysis under the axioms of ZF has been developed to invent a version of Banach-Tarski which is independent of AC.
What does the Banach-Tarski paradox look like without AC? Are there any versions of it? (For an example, one of the theorems that has been proven without AC is the Heine-Borel theorem.)

REPLY [57 votes]: According to Dougherty and Foreman's 1992 PNAS paper, Banach-Tarski paradox using pieces with the property of Baire (doi:10.1073/pnas.89.22.10726), the following result can be shown without AC: the unit ball has a finite collection of disjoint open subsets that transforms by rigid motions to another collection of disjoint open sets such that the closure of the union is two unit balls.