Sets, the Axiom of Choice, And All That: A Tutorial

This tutorial deals with the application of the Axiom of Choice in one of its popular disguises to objects which are of some interest in computer science (like lattices, Boolean algebras, filters and ideals, games). We discuss some common variants of this axiom such as Zorn's Lemma, Tuckey's Maximality Principle, the Well-Ordering Theorem and the Maximal Ideal Theorem; each equivalence gives applications its due attention. We show that the Axiom of Choice can be used to demonstrate the existence of non-measurable sets in the real line. This is an occasion to introduce some measure theory within the context of Boolen σ-algebras. Games are introduced as well, and the Axiom of Determinacy is discussed, giving rise to show that this axiom can be used to demonstrate that each subset of the real line is measurable. Hence we use games as a tool for proofs. We try to shed some light on the slightly complicated and irritating interplay between these two axioms. We assume the basic knowledge of mathematics that is introduced by a one year course for beginning computer scientists at a German university. A grain of mathematical maturity may help as well. Some exercises are offered, and solutions are suggested as well.