The axiom of choice has gotten its bad reputation because it leads to contradictions with uncountable sets. But it is a very natural axiom and is frequently applied without notice in mathematics as Zermelo has correctly pointed out when defending his invention against objections of Borel, Peano, Poincaré, and others. [E. Zermelo: "Neuer Beweis für die Möglichkeit einer Wohlordnung", Math. Ann. 65 (1908) pp. 107-128]
The problem is only, as mentioned above, the application of the axiom to uncountable sets. In the history of mathematics, it was usual to fix factual conventions by an axiom, like "it is possible to draw a straight line from any point to any point" or "if n is a natural number, then n+1 is a natural number". The application of the axiom of choice claims for the first time a counterfactual convention, namely to choose an element without knowing what is chosen.
At least in 1904 it was clear that there are only countably many finite strings of letters, including strings defining mathematical objects. Cantor knew this theorem, as he wrote in a letter to Hilbert in 1906, although he did not believe that it is true. "If König's theorem was true, according to which all 'finitely definable' real numbers form a set of cardinality aleph_0, this would imply that the whole continuum was countable, and that is certainly false." [G. Cantor, letter to D. Hilbert (8 Aug 1906)]
Today there is no doubt that König's theorem is true. In order to maintain transfinite set theory, it is necessary to have the (in this realm) counterfactual axiom of choice for proving the basic theorem of set theory: Every set can be well-ordered. Otherwise a lot of ordinal theory would be unprovable. Therefore set theorists have agreed that the axiom is "not constructive", i.e., we can prove that we can choose every element, but we cannot choose every element. Although Zermelo used the axiom to prove that every set can be well-ordered, i.e., he thought it could be done and not only be proven that it could be done, knowing that in fact it cannot be done. [E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904)]
But what is the value of a counterfactual axiom? We could state many other axioms of same value like:
Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)
Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)
Axiom of prime number triples: There is a second triple of prime numbers, besides (3, 5, 7). (But provably this second triple is not arithmetically definable.)
Axiom of meagre sum: There is a set of n different positive natural numbers with sum n*n/2. (This axiom is not constructive. Provably no such set can be constructed.)
All theories based upon such axioms would have the same value as transfinite set theory, namely none.
Keeping this in mind and ignoring absurd attempts to apply uncountable alphabets or infinite definitions to define uncountably many elements, we can be sure that the axiom of choice is true in every world with correct mathematics and therefore without uncountable sets.