Paradoxical decompositions of free F2 -sets and the Hahn-Banach axiom
Résumé
Denoting by F2 the free group over a two-element alphabet, we show in set-theory without the axiom of choice ZF that the existence of a (2, 2)-paradoxical decomposition of free F2 -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
ZF 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 R of real numbers. Our result is linked to the (m, n)-paradoxical decompositions of free F2 -sets previously obtained by Pawlikowski (m = n = 3, see [11]) and then by Sato and Shioya (m = 3 and n = 2, see [13]) with the sole Hahn-Banach axiom.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)