First a quick question regarding the definition of the axiom of choice. Do the sets have to be mutually disjoint nonempty sets or just non-empty? One source states: "For any set X of nonempty sets, there exists a choice function f defined on X." But another source states that the sets have to be mutually disjoint.
Secondly, pardon me if I sound ignorant (I'm learning this as a hobby so I don't have much background or time for it) but isn't it a really obvious/self-evident concept? I mean essentially, it is saying that if you have a collection of non-empty sets, then you can pick an element out of each set. I realize that there are difficulties when we cannot make explicit choices because we cannot create an explicit algorithm for the choice function (for example the collection of all nonempty subsets of the real line), but does that really matter?
I mean just like the number 5, the existence of the function 'f' is purely formal. Math isn't able to fully describe or prove everything but doesn't mean it doesn't exixt.