Emil Båth: Axiom of Choice

Abstract:

This thesis investigates how the Axiom of Choice (AC) is used in mathematics. Some weaker forms of AC are presented and proved to follow from AC. The Well-Ordering Theorem, Zorn’s Lemma and the Fundamental Theorem of Linear Algebra are proven to be equivalent to AC. Two consequences of AC are shown; the useful Boolean Prime Ideal Theorem and the counter-intuitive Banach-Tarski Paradox.