The axiom of choice asserts that any nonempty product of nonempty sets is nonempty. This seems so trite, it is amazing that there should be any controversy at all in accepting it. It seems unnecessary to state as an axiom, it ought to follow from the other axioms of set theory; but it doesn't, and has many surprising counter-intuitive consequences. If set theorists are interested in models of set theory in which choice fails, it is for good reasonyou can prove theorems by starting with models in which strong combinatorial principles hold that negate choice. Some of these (e.g. the axiom of determinacy of games) are interesting in themselves. Further, given strong enough assumptions about large cardinals, there are natural models in which choice fails spectacularly. So the advice isdo assume the axiom of choice, but do not be complacent about it. The talk is a basic introduction (and invitation) to this fascinating world.
