Paradoxical decompositions of free F2F₂‐sets and the Hahn‐Banach axiom

Denoting by the free group over a two‐element alphabet, we show in set‐theory without the axiom of choice that the existence of a (2, 2)‐paradoxical decomposition of free ‐sets follows from the conjunction of a weakened consequence of the Hahn‐Banach axiom and a weakened consequence of the axiom of choice for pairs. The existence in of a paradoxical decomposition with 4 pieces of the sphere in the 3‐dimensional euclidean space follows from the same two statements restricted to the set of real numbers. Our result is linked to the ‐paradoxical decompositions of free ‐sets previously obtained by Pawlikowski (, cf. [11]) and then by Sato and Shioya ( and , cf. [13]) with the sole Hahn‐Banach axiom.