The Axiom of Choice and Other Fallacies

The full Axiom of Choice states that given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. One easy (and not quite right) way of thinking of it is that given a set, you can always pick an element of it. Sound obvious, huh?

Well... not quite. The problem comes when the set becomes infinite. Then how you tell me what you have picked becomes an issue. If the set is only countably big, so that you can number the elements 1, 2, 3 and so on, it isn't an issue - you say `I pick element 14,784,...' or whatever. But if the set is bigger than that, for instance it has the same cardinality as the real line, then specifying what you have picked gets harder.

In particular, set theorists have proved adept at constructing very large sets indeed - the boundary of `stupidly big' starts somewhere around the totally ineffable cardinals. For these babies, specifying the choice that you have made requires so much information that some mathematicians reject the axiom of choice as not effective. Basically they think that if you can't say what you have chosen without ridiculous amounts of information, then you can't choose. It also turned out that full AOC was equivalent to other principles that people found troubling, such as the law of the excluded middle. Some mathematicians therefore rejected full AOC, accepting only the axiom of choice when applied to `reasonable small' sets. (Thus we get for instance realizable versions of AOC, where you can apply AOC to `nice' sets.)

So what, economics lovers? Well, it turns out if Chris Ayers is to be believed that lots of economics relies on AOC

All current solution concepts in game theory also require the theorems implied by AC. In particular, lexicographic utility, lexicographic probability, the real line being well-ordered, and the existence of a universal space are all equivalent to AC; therefore any argument to disprove their existence must be false. Any proofs using properties that fail under AC must be redone. The concept of Nash Equilibrium becomes either a tautology (in the absence of AC) or violates rationality (in the presence of AC); we provide an example demonstrating this.

My strong suspicion is that this is a storm in a teacup and that even if Ayers' result is true (which it may well not be - caveat lector), you can get by with a weaker `effective' version of AC by considering suitable realizable outcomes. You'd end up with a smaller collection of games, but this would probably include realizable versions of all of the interesting ones. Still, even if this is nonsense, the idea of setting up game theory in a more effective setting is interesting.

Update. A cursory search doesn't reveal any academic association for Mr. Ayers. And the proof of the first theorem is wrong. That's not a proof that Ayers is a charlatan, of course, but should make one more sensitive to the possibility.