axiom of choice

Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and there is no "rule" for which object to pick from each. The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. (Source)

Axiom of Choice is controversial because it leads to some weird theorems such as the following:

The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points... Unlike most theorems in geometry, this result depends in a critical way on the axiom of choice in set theory. This axiom allows for the construction of nonmeasurable sets, collections of points that do not have a volume in the ordinary sense and require an uncountably infinite number of arbitrary choices to specify... The existence of nonmeasurable sets, such as those in the Banach–Tarski paradox, has been used as an argument against the axiom of choice. Nevertheless, most mathematicians are willing to tolerate the existence of nonmeasurable sets, given that the axiom of choice has many other mathematically useful consequences... It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another. (Source).

This counter-intuitive result is partly due the bizarreness of set theory itself. Our intuition is based on classical physics. However the category of sets has nothing remotely to do with physics. It has the least structured morphisms possible. Unlike their set theoretical counterparts, "physical points" have cohesive relationships. Therefore, unlike set theoretical functions, physical transformations exhibit a substantial amount of structure. In particular, they can not perform the reassembly process required in the Banach-Tarski paradox, because moving a complicated subset of a solid ball without affecting the other subsets is physically impossible.

Of course it is the Axiom of Choice that allows one to select the pathological subsets in the first place. In other words it is the Axiom of Choice that is the true source of the paradox. However the paradox feeling arises only when the selected subsets are spatially reassembled into two identical copies of the original ball.

(At a deeper level the real culprit is the Space-as-a-Set-of-Points metaphor. This mathematical metaphor, like any other metaphor, does not capture exactly our intuitive notion of space. The paradox feeling arises from this deficiency.)