The Axiom of Choice

Fun thing to think about for today: The Axiom of Choice. Math is a lot like Perry Mason - he gets you in the witness stand, and asks you if you like chocolate. It goes like this

1. You answer.
2. ???
3. You confess to the murder: "I'm not stupid! Could a dumb person have killed her like I did?!"

We are the Hamilton Burgers of reality [1]. Again, it starts innocently enough:

Let X be a set of non-empty sets. Then we can choose a single member from each set in X.

Which sounds fairly dry, but hang with me here. For one thing, the axiom says nothing about how to choose each element; It only says that it is possible to do so. It's easy to see this with finite sets or well-ordered sets (e.g., the positive integers), but the axiom also applies to sets like the real numbers, which are uncountably infinite and not well ordered (see the article for details) . This means that we accept the existence of a function, called the choice function, that we may have no idea of how to implement. And so?

One reason that some mathematicians dislike the axiom of choice is that it implies the existence of some bizarre counter-intuitive objects. An example of this is the Banach–Tarski paradox which says in effect that it is possible to "carve up" the 3-dimensional solid unit ball into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell us how to carve up the unit sphere to make this happen, it simply tells us that it can be done.

Which would be a nice trick, because you wouldn't conserve mass, for instance. The paradox is generally thought to resolve itself because to it's not physically possible to cut a real, atomic object into the necessary pieces - but, it may not be as reassuring as you would hope:

At first glance, the Banach-Tarski result seems to contradict some of our intuition about physics -- e.g., the Law of Conservation of Mass, from classical Newtonian physics. If we assume that the ball has a uniform density, then the Banach-Tarski Paradox seems to say that we can disassemble a one-kilogram ball into pieces and rearrange them to get two one-kilogram balls. But actually, the contradiction can be explained away: Only a set with a defined volume can have a defined mass. A "volume" can be defined for many subsets of R³ --- spheres, cubes, cones, icosahedrons, etc. --- and in fact a "volume" can be defined for nearly any subset of R³ that we can think of. This leads beginners to expect that the notion of "volume" is applicable to every subset of R³. But it's not. In particular, the pieces in the Banach-Tarski decomposition are sets whose volumes cannot be defined.

Which is certainly good news for my investment in Krugerrands [2]... But it still unsettling that this paradox exists. But what about it's negation? Quoting from the Wikipedia article again:

On the other hand, the negation of the axiom of choice is also bizarre. For example, the statement that for any two sets S and T, the cardinality of S is less than or equal to the cardinality of T or the cardinality of T is less than or equal to the cardinality of S is equivalent to the axiom of choice. Put differently, if the axiom of choice is false, then there are sets S and T of incomparable size: neither can be mapped in a one-to-one fashion onto a subset of the other.

This is somewhat more than a curiosity too because many fundamental results are derived using this axiom. You can take an agnostic approach, and that's what many mathematicians choose to do (not unlike quantum mechanics), but it results in that many questions are then undecidable. Undecidable questions are okay, because at the basis of what we consider to be common sesible are many such statements that we accept as true and then move forward to derive more complicated results (trivial and non-trivial[3]). But it seems strange here that even if you assume the statement, either way you have paradoxical results.
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[1] Random thought - which is more futile: Burger's 10-year prosecutor's losing streak, the Washington Generals million game losing streak versus the Harlem Globetrotters, or the Nazi's on the History Channel?
[2] Actually my long term investment plan is to buy lottery tickets where the take-home value of the jackpot divided by the odds of winning is greater than the cost of a ticket. It could happen.
[3] Any result you've derived is trivial; Anything I haven't is non-trivial.