TITLE: Is there a *simple* example of how the axiom of choice can lead to a counterintuitive result?
QUESTION [27 upvotes]: This question is quite subjective, but here goes:
The axiom of choice notoriously leads to many extremely counterintuitive results, like the Hausdorff, Von Neumann, and (most famously) Banach-Tarski paradoxes. Unfortunately, the proof of these paradoxes is also notoriously complicated, and it's not at all obvious why an axiom as seemingly innocuous (and, in fact, obvious) as the axiom of choice has such strange consequences.
Do there exist any examples of a simple chain of reasoning that leads from the axiom of choice to a counterintuitive result? (Though not necessarily a result as spectacularly counterintuitive as the Banach-Tarski paradox.) One which is simple enough that someone without training in advanced mathematics could understand the general train of logic (with perhaps a few steps fudged over)? I've never see an example that lets me intuitively understand why naive intuition "goes wrong" exactly at the step where we use the axiom of choice.
(To be clear: I'm not looking for a simple counterintuitive result that follows from the axiom of choice - Banach-Tarksi would qualify for that. I'm instead looking for a counterintuitive result that follows from the axiom of choice with a (relatively) simple proof.)

REPLY [2 votes]: I'd say the easiest example would be the construction of Vitali sets.  I expect you already understand this, but here's how I might explain it to a 12th-grader.
Start with the naive concept of length, beginning with that of an interval, and the common notions that breaking up or shifting an interval doesn't change its length, that the lengths of a finite collection of disjoint intervals add, and that this extends to a countable sequence of disjoint intervals when its sum converges.  Stress here that the real numbers are not countable.  Point out that, since all of these lengths are nonnegative, the terms of such a sequence can be rearranged freely, and that therefore all this holds for any set of real numbers that can be built up from a countable collection of intervals.  From this quietly suggest, as intuition might dictate, that this can be extended to any set of reals.  All of this is intuitive enough.
Now, point out that two irrational numbers having a rational difference is an equivalence relation, and so produces equivalence classes.  Consider these equivalence classes restricted to $[0,1)$.  Apply the axiom of choice to get a set of one number from each such equivalence class.  Point out that shifting it by a rational number, cutting off the part outside the interval, and shifting it back to the other end will create a set that should have the same length.  Show that applying this for all the rational numbers in that interval will generate a partition.
So now you have a set of length one partitioned into a countably infinite collection of sets of the same length.  That length can't be zero, since the collection is countable, but it also can't be more than zero, since that would give their union infinite length.  Therefore, the axiom of choice allows you to construct sets of real numbers that do not have a length.