Zermelo’s Axiom Of Choice: Its Origins, Development and History by Gregory H Moore
The axiom of choice is a fascinating subject which is used in most undergraduate mathematics classes as a means for justifying Zorn’s lemma which is in turn used to prove a number of theorems. I remember finding it really confusing when I first heard the witty remark (by Russell I think) “given an infinite set of pairs of shoes it’s easy to choose a member of each pair, but not for an infinite set of pairs of socks”.
This book covers the history of the axiom. Zermelo formulated his well ordering principle around the time that mathematicians were investigating set theory and its application to the foundation of mathematics. It took a long time before people accepted that there was a need for infinite choices in a mathematical proof and there were many discussions about whether such a principle was needed in a given proof. Before the formulation of first order logic, and its use for formulating set theory, fairly famous mathematicians were on opposing sides during the debate, and the book covers this social aspect of mathematical discovery.
A fairly long read which is a little dry in places, but a very interesting read. As an appendix it lists loads of famous theorems and whether they are equivalent or are weaker with and without the use of the axiom of choice.