Paradox Lost and Paradox Regained: The Banach-Tarski Paradox and Ancient Paradoxes of Infinity

The Banach-Tarski Paradox (BTP) is often described as a proof that spheres in R3 can be doubled by the isometric action of rotations on a finite number of their parts. This dissertation examines the BTP through its relationship to the Axiom of Choice (AC) and its similarity to ancient paradoxes of infinity whereby magnitudes are doubled. I start by examining the relationship between infinity and AC and the mathematical relationship of double-able size to infinity via the group-theoretic notion of 'paradoxical'. After moving through the proof of the BTP, AC is given an intuitive explication, which is supported by its comparison to a permutation model for which AC does not hold analogous to Russell's metaphorical socks. Subsequently, I compare and contrast two sets of doubling paradoxes owing to Zeno and the atomists Democritus, Epicurus and Lucretius. Both sets are shown to rely the attribution of properties of indivisibles – as exemplified by Aristotelian 'units' – to intrinsically divisible magnitudes. Combining the conceptualizations of AC and the ancient paradoxes, I conclude that the BTP hinges upon a similar mismatching of properties as in the ancient paradoxes but in a subtler and less tractable way for Real measure.