The Banach-Tarski Paradox and Weakenings of the Axiom of Choice

Abstract

Since its discovery, the Banach-Tarski Paradox has been a source of puzzlement and has led to much questioning of the Axiom of Choice. In this thesis, we discuss the paradox itself as well as its relation to critical weakenings of choice in the pursuit of understanding how much of mathematics and logic must be invalidated to get rid of the paradox. In particular, the Weak Ultrafilter Theorem and the Hahn-Banach Theorem are shown to each still lead to the Banach-Tarski Paradox. This thesis includes the exposition of previous findings regarding relationships between these theorems and the Banach-Tarski Paradox, as well as novel proofs of some of the previously unknown relationships. We demonstrate that these weakenings of choice still fail to eliminate the Banach-Tarski Paradox while also rendering mathematical results critical to the development of measure theory and analysis unprovable. While the paradox violates our most basic mathematical intuition, mathematics without it is perhaps even more nonintuitive. In conclusion, mathematicians must continue to question the axioms of set theory in order to fully understand the effects of controversial axioms and paradoxical results, all in pursuit of finding the most fitting foundation for the development of mathematics.