The Axiom of Choice is Wrong (2007)

The problem solution in the article uses the axiom of choice to construct a "nonprincipal ultrafilter" on the natural numbers. This is actually weaker than the full axiom of choice, but you can still show that no such object is computable. It's a nice exercise to show that with the same assumptions as in the article you can decide the halting problem. (hint: consider the boolean sequence where the nth element is true iff the Turing machine halts within n…

One of the troubling aspects of the Axiom of Choice is that both affirming and denying it results in some pretty unintuitive implications (but different ones). This MathOverflow discussion has some interesting examples of unintuitive implications of the axiom, along with a bonus answer giving examples of unintuitive implications of not having the axiom: http://mathoverflow.net/questions/20882/most-unintuitive-app... The history of it is full of people wrestling with both sides of that, including many participants…

So this guy is against using the axiom of choice because a countably infinite number of prisoners in a row who have agreed on an infinite set of equivalence classes just between breakfast and hat game time and remember them and each of them can recognize infinitely many hat colors in finite time (takes deep breath) suddenly becomes unintuitive using the axiom of choice? I can't help him there.

A related, and to my mind even more counter-intuitive result, is a strategy for predicting f(x) given only the values {f(y), y<x}, where f is any arbitrary function on the reals. https://www.math.upenn.edu/~ted/203S10/References/peculiar.p... [PDF]