Using a well-ordering on the reals, one can prove there exists a partition of the three-dimensional Euclidean space into unit circles (PUC). We show that the converse does not hold: there exist models of $\mathsf{ ZF}$ without a well-ordering of the reals in which such partition exists. Specifically, we prove that the Cohen model has a PUC and construct a model satisfying $\mathsf{ DC}$ where this is also the case. Furthermore, we present a general framework for constructing similar models for other paradoxical sets, under some conditions of extendability and amalgamation.
Categories: math.LO math.MG