lo.logic - Most 'unintuitive' application of the Axiom of Choice?

The most destructive aspect of uncountable choice is that it conflicts with random choice. With uncountable choice, any object which is constructed using randomness, like a random walk, a random field, or even a randomly picked real number, cannot exist, because there are sets which it cannot consistently be assigned membership to.

In order to define what it means to have a random walk, or a random graph, or a random infinite Ising model configuration, or whatever, you need to define what it means to have an infinite sequence of random coin flips. The result can be encoded as a real number, the binary digits of which are the results of the coin flip, and if this real number really exists, as an actual mathematical object, then this object either belongs to any given set S, or it doesn't.

It is so intuitive to think of random objects this way, that they are often illustrated with pictures, showing us what they look like (see http://en.wikipedia.org/wiki/Wiener_process for a picture of a "realization" of a random walk). These pictures do not signify anything when the axiom of choice is present.

The reason is that once you have actual random objects, for which you can assign membership to any set S, then you can define the probability of landing in S by choosing random objects again and again, and asking what fraction of the time you land in S. This always converges, because given any long finite sequence of 1's and 0's which represent independent random events, any permutation of the 1's and 0's has the same likelihood. This means that it is probability 0 that the seqeunce will oscillate in any way, and with certainty it will converge to a unique answer.

This answer is the measure of the set S, and every set is measurable in this universe. This makes analysis much easier, because everything is integrable, measurable, etc. This is so intuitive, that if you look at any probability paper, they will illustrate with random objects without hesistation, implicitly denying choice.

(I realize that this answer overlaps with a previous one, but it corrects a serious central mistake in the former.)