 The Axiom of Choice (AC) is a philosophical puzzle in the foundations of mathematics. Roughly speaking, this axiom, that originally emerged in set theory, says that if we have a set \(A\) of non-empty disjoint sets, then we can form a new set \(B\) which contains exactly one element from each set in \(A\). What can be seen as a problem is the fact that when \(A\) is infinite, forming the set \(B\) seems to require making an infinite number of choices. See entry on  Axiom of Choice  for a thorough definition and discussion. On the one hand the Axiom of Choice seems magic, for example when it implies that the wildest sets have a well-ordering. On the other hand it seems innocuous, for example when it implies that the Cartesian product of a family of non-empty sets is non-empty. The Axiom of Choice is typically used to construct odd “paradoxical” sets, such as a non-Lebesgue measurable set of real numbers, a well-ordering of the real numbers, a paradoxical decomposition of the sphere (see entry on  set theory),  and so on. Such sets are usually in one sense or another undefinable. For example, a non-Lebesgue measurable set of real numbers has to be undefinable in second-order logic over \((\oN,+,\cdot)\), if there are sufficiently large cardinals (see the entry on  large cardinals and determinacy).