Axiom of Choice, Non-Measurable Sets, Amenability and the Banach-Tarski Paradox

This manuscript presents a standalone, structurally unified analysis connecting the Axiom of Choice, non-measurable sets, amenability, and the Banach-Tarski paradox. Rather than treating these as isolated mathematical anomalies, the paper demonstrates how they form a strict, logical chain across set theory, measure theory, and geometric group theory. The framework first isolates the selector mechanism supplied by the Axiom of Choice, showing how it produces quotient representatives for dense equivalence relations. This directly enables the construction of Vitali sets, yielding a mathematical proof that no countably additive, translation-invariant measure can be extended to all subsets of the real numbers. The manuscript then shifts to group actions, developing the concept of paradoxical decomposition and establishing amenability as the exact algebraic boundary that obstructs it. By proving the non-amenability of the free group on two generators and embedding it into the 3D rotation group, the Banach-Tarski paradox is derived as a natural geometric consequence. Conversely, the failure of the paradox in 2D is explicitly explained through the amenability of the planar Euclidean isometry group. Finally, the entire phenomenon is placed in its proper logical context by discussing Solovay's model, clarifying that these paradoxes are not geometric inconsistencies, but the inevitable structural result of combining unrestricted global selection with infinite orbit structures and non-amenable symmetry.