How to get rich without the axiom of choice

This is the title of a talk given at the CMU Graduate Student Seminar on February 23, 2016. A video of the talk is on Youtube (if you don't like Youtube, here is a direct link).


The Banach-Tarski paradox says that one can decompose a unit ball into finitely many pieces, and move and rotate these pieces to obtain two disjoint unit balls. One can use this as a "get rich quick" scheme: buy one unit ball of gold, duplicate it using Banach-Tarski, sell it back, and repeat.

One may consider the paradox (or our failure to get rich by using it) as evidence that the axiom of choice is false. In this talk, I will assume that choice fails and discuss another paradox: the existence of an equivalence relation with more classes than points. This leads to another "get rich quick" scheme: get (or borrow) as much money as there are real numbers, put your money in boxes in a smart way, then sell each box for a dollar. The argument I will give is based on slides by Michael Ray Oliver titled "How to have more things by forgetting where you put them".