Multiple context available for the AC? I am surprised at the language used in connection with the axiom of choice. From the answer to a question a made (which turned out to be duplicate) about involvement of AC in Wiles’ proof of Fermat’s last theorem I note that the axiom (AC) is treated like a conjecture that (according to some) might be - or obviously is (according to some) – true. 
My question is: In what sense could an axiom be true if it is independent of the Peano axioms (supplemented with addition and multiplication), or are there several contexts available in which the axiom may be true in one and not (ed: or unprovable) in another?    
 A: The axiom of choice have nothing to do with number theory an Peano axioms. The language in which the axiom of choice is formalized has nothing in common with the language in which the Peano axioms and number theory are written.
However, when proving a statement one sometimes uses set theory as a formal theory in the background (even without referring to it explicitly. Simply by referring to sets, models, and so on), such theory is called "meta-theory", as say that we work "in that theory". The axiom of choice is a part of our meta-theory.
One can ask, if so, do we need the axiom of choice as part of our meta-theory for a certain proof to hold? Sometimes the answer is yes. For example if one wants to prove that every field has an algebraic closure. But it turns out that for statements about the natural numbers which are "simple enough", e.g. first-order statements in the language of Peano axioms, the axiom of choice does not enhance our power to prove things. So if we proved Fermat's last theorem with $\sf ZFC$ as a meta-theory, we can in fact prove it within $\sf ZF$.
(One caveat is when proving statement in set theory, one can talk about the axiom choice in the theory, in which case one proves a statement in the language of the theory, with the axioms of the theory which may or may not include the statement known as "the axiom of choice".)
