The Axiom of Choice by Thomas J. Lech
Ever since I did the course on Set Theory and Logic at university, I have found the Axiom of Choice quite fascinating. Applications of the axiom of choice, mostly via Zorn’s lemma, are found all over the place in undergraduate mathematics, where it is pointed out that the axiom is sometimes too powerful and leads to a number of undesirable results. In particular, the existence of unmeasurable sets of real numbers and the decomposition of the sphere into two spheres of the same size (using a physically unrealisable construction). The introduction to the book covers these undesirable results.
Chapter two looks at the good consequences of the axiom, and chapter three looks at what it means for an axiom system to be consistent, discussing models and relative consistency. Chapter four introduces set theory extended with atoms, items that don’t have members, and shows that in this theory the axiom of choice is unprovable and is independent of the ordering principle.
Chapter five looks at generic models and forcing to demonstrate the independence of the Axiom of Choice – the technique of forcing is what allowed Paul Cohen to prove the independence of the Continuum Hypothesis in the 1960s. Chapter six looks at embedding, showing how consistency of ZF can be moved to the construction of various permutation models. There are other chapters on various independence results, mathematics without the axiom of choice and a look at some properties that contradict the axiom of choice.
The book is obviously quite technical, but has good high level explanations of how the proof is going to flow, and the chapters end with a discussion of the importance of the various results and their historical setting. If you find set theory and logic interesting, it is well worth a look.