The article leans pretty hard on the hat problem to show that the Axiom of Choice has intuitively unacceptable consequences, but I don't see why the solution is so "ridiculous," as they put it. Here's one "intuitive" way of looking at the problem: every person in the line sees an infinite number of hats in front of them, but not the finite number behind them. Intuitively, they see 100% of the information in the problem. The amount of information each of them can see is infinite, and the amount they can't see is finite. Is it so strange then that they only make a finite number of wrong guesses, and an infinite number of correct ones?

I think the article artfully uses a framing of the problem that hides the inherent, inexorable alienness of infinite sets under a familiar-seeming surface just long enough for it to come bursting out in the solution. It's a trick of timing. If your brain lingers on the problem long enough, you realize that the problem itself violates our ordinary intuitions, independent of the solution. The idea of an infinite number of people agreeing on and memorizing representative members of R^N from each equivalence class and then determining the relevant equivalence class by reading an infinite number of real numbers from an infinite number of hats is completely fantastical. How could any solution to this problem not be nuts?

More to the point, if the problem is presented without a bonkers framing that already violates our intuition, is the solution from the Axiom of Choice still counterintuitive? I didn't find it to be so when I first encountered it in class. Maybe the intuitiveness depends entirely on the metaphor used to bring intuition to bear.