Honours Thesis: ZFC and the Banach–Tarski Paradox

Introduction

As part of the requirements of my undergraduate degree, I completed an Honours Seminar course. I had to select an area of mathematics with which I didn't have much experience and work with a member of the faculty to complete a written thesis and give a 20-minute talk. Under the supervision of Dr. Andrew J. Dean, I learned about Zermelo–Fraenkel set theory, the Axiom of Choice, and the Banach–Tarski paradox. I have long been interested in the foundations of mathematics—both the history and the philosophy of mathematics appeal to me, and I enjoyed this opportunity to apply my knowledge of propositional logic as I learned about axiomatic set theory. My thesis is an overview of the history of ZFC, from Cantor's naive set theory to the modern presentation of the axioms. It culminates with a presentation of the proof of the Banach–Tarski paradox.

At 19 pages, my thesis is slightly too long to be worth typesetting for the Web for now. I make both it and the slides I used for my talk available under the usual Creative Commons Attribution License.