Simon Almerström Przybyl: Choice Principles in Mathematics

In this thesis we illustrate how mathematics is affected by the Axiom of Choice (AC). We also investigate how other choice principles affect mathematics. Proofs of the following three major results are presented:

• AC, Zorn's Lemma and the Well-Ordering Theorem are equivalent. We prove this equivalence without using transfinite techniques.
• The Banach-Tarski Paradox (BTP) holds in ZFC but fails in ZF + AD + DC and is thus independent of ZF + DC. The latter results are proved under certain consistency assumptions using the connection between BTP and non-measurable sets.
• AC and Tychonoff's Theorem are equivalent.

Proofs of other minor results regarding choice principles are also presented.