Page "Axiom of choice" Paragraph 33

Another argument against the axiom of choice is that it implies the existence of counterintuitive objects.

One example is the Banach – Tarski paradox which says that it is possible to decompose (" carve up ") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.

The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets.