First: My apologies for the probably imprecise way I am phrasing this question, to the point where I expect I will be told it is actually meaningless as presented here.
The axiom of choice can be restated as 'the Cartesian product of a collection of non-empty sets is non-empty'.
But '... non-empty' seems to only ensure that at least one element from each set is selected in the process, while usually, when talking about Cartesian products, we seem to expect that all elements are eventually selected in the process, e.g. the Cartesian product of sets A, B is the set of all ordered pairs (a, b) s.t. a is in A, b is in B.
My question is then: is there a "stronger" form of the axiom of choice, that ensures that each element in each set of a family of sets is selected? And if that would be in fact a stronger requirement: is there any mathematical use or motivation for such a stronger version of the axiom?
In a probably related question to my own, I found the following reply:
But what about Cartesian product of three sets? ... What objects do you have when you take the product of infinitely many sets? ... it turns out that the Cartesian product of sets is actually the set of choice functions from them. The axiom of choice assures that every family of non-empty sets has a choice function, and therefore the Cartesian product of any family of non-empty sets is non-empty
If I understand it correctly, in order to even ask a meaningful question, I need to reformulate as follows:
The axiom of choice assures that every family of non-empty sets has a choice function. Is there a (non-paradoxical) way to speak about a set of all possible choice functions, given any family of non-empty sets? If so, would that be a stronger version of the axiom of choice? If so, would there be any use for it?