Paradoxical decompositions of free F2$F_2$‐sets and the Hahn‐Banach axiom

• Published in: Mathematical logic quarterly Vol. 70; no. 4; pp. 367 - 387
• Main Author: Morillon, Marianne
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.11.2024
• Subjects: Decomposition; Euclidean geometry; Real numbers
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202400003

Denoting by F2$F_2$ the free group over a two‐element alphabet, we show in set‐theory without the axiom of choice ZF$\mathsf {ZF}$ that the existence of a (2, 2)‐paradoxical decomposition of free F2$F_2$‐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$\mathsf {ZF}$ 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$\mathbb {R}$ of real numbers. Our result is linked to the (m,n)$(m,n)$‐paradoxical decompositions of free F2$F_2$‐sets previously obtained by Pawlikowski (m=n=3$m=n=3$, cf. [11]) and then by Sato and Shioya (m=3$m=3$ and n=2$n=2$, cf. [13]) with the sole Hahn‐Banach axiom.