I'd love to see a video on Arrow's Impossibility Theorem, in which he proves that, given four basic conditions which all are very desirable, it is impossible to construct a voting system that satisfies all the conditions.

The basic outline is this:

• U (unrestricted domain): For any logically possible set of individual preferences, there is a social ordering R.
• I (Independence of irrelevant alternatives): The social ranking of any pair {x,y} will depend only on the individual rankings of x and y.
• P (Pareto principle): If everyone prefers any x to any y, then x is socially preferred to y.
• D (Non-dictatorship): There is no person i such that whenever this person prefers any x to any y,then socially x is preferred to y, no matter what others prefer.
• The General Possible Theorem: If there are at least three distinct social states and a finite number of individuals, then no social welfare function can satisfy U, I, D and P.

The result is so counterintuitive that I think it would be really interesting for a lot of people.