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Preprints, Working Papers, ... Year : 2024

Paradoxical decompositions of free F2 -sets and the Hahn-Banach axiom

Abstract

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.
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Dates and versions

hal-04540612 , version 1 (10-04-2024)

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  • HAL Id : hal-04540612 , version 1

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Marianne Morillon. Paradoxical decompositions of free F2 -sets and the Hahn-Banach axiom. 2024. ⟨hal-04540612⟩
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