Cool Logic

Tanmay Inamdar (ILLC)

Horrors with and without the Axiom of Choice: With Particular Emphasis on a Proof of the Banach-Tarski Theorem

October 19th at 17:30, in Science Park D1.113

In my talk I will talk about things that can go wrong with and without the Axiom of Choice. As a warm up, I will show how AC allows us to construct a non-measurable set of Reals.
 Next, I will give a proof of the Banach-Tarski theorem, which is one of the few theorems in mathematics that has an appeal beyond mathematics. While many of us may have laughed at xkcd jokes referring to it, its proof is something that even someone with an undergraduate degree in mathematics might not have seen. Surprisingly, the proof is not too hard, and anyone who has some geometric intuition (and a little trust in me) should be able to follow the steps.

Despite its unintuitive consequences, the Axiom of Choice is still accepted by the majority of mathematicians. In the last part of the talk, I will try to explain why, by giving some examples of things that may go wrong when AC is absent.

I will not be assuming too much mathematical background on the part of the audience. Knowing the definitions of a group and a group action would be helpful, but I will give them as well, and draw some pictures to motivate them. No knowledge of measure theory is required; basic definitions will be motivated, and the non-measurable set will fall out easily. Geometric intuition (e.g. a sphere being rotated) would be the most useful prerequisite, but I will draw as many pictures as I can to help people along. No knowledge of set theory is required.

Other benefits of attending are that you get to see the Hilbert Hotel argument being used in a mathematical proof, and the next time you read a webcomic which jokes about the Banach-Tarski theorem you can feel smug about not only knowing what the joke is, but why the joke works. You will also get to see the first proof in this new edition of Cool Logic.