Partitions of ℝ³ into unit circles with no well-ordering of the reals

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 Z F \mathsf {ZF} without a well-ordering of the reals in which such partition exists. Specifically, we prove that the Cohen–Halpern–Lévy model has a PUC and construct a model satisfying D C \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 .