The axiom isn't exactly about being able to pick an arbitrary element out of an arbitrary set. It says that if you have a collection of nonempty sets, then the

The axiom isn't exactly about being able to pick an arbitrary element out of an arbitrary set. It says that if you have a collection of nonempty sets, then there is a function f from the collection X of sets to the union of the collection of sets, with f(A) an element of A, for A in X. To say each A is nonempty is kind of already saying that we can pick out an element from each set. The axiom just goes on to say that there is an actual function which does the picking out. I think of it as taking the meta-level (a statement that every set in a collection of sets is nonempty) to the set level (there exists a function which picks out one of those elements for each set in the collection). The axiom doesn't actually say what the function is. It just says that the set of functions with the given picking-out property is nonempty. A similar statement to the axiom is that every sequence of nonempty sets A_1,A_2,... has at least one sequence a_1,a_2,... with a_i in A_i. A consequence of the axiom of choice is that every vector space has at least one basis. I'm not sure whether every vector space having a basis is equivalent to the axiom of choice, though.