Simple question: Which is the Wikipedia definition of axiom of choice

Question

I looked up Wikipedia, and on the top of the page it says:

For every indexed family $(S_i)_{i \in I}$ of nonempty sets, there exists an indexed family $(x_i)_{i \in I}$ of elements such that $x_i \in S_i$, for every $i\in I$

Then when I scroll down a bit, Wikipedia switches the definition to:

For any set $X$ of nonempty sets, there exists a choice function $f$ defined on $X$

The top definition seems to be easy and intuitive. Why obfusticate the picture with this new object called choice function?

Which of these should be considered the "working" definition of Axiom of Choice. When writing a math proof, or talking to a random stranger on the street, which statement of axiom of choice is preferable?

More concretely, you are writing a math paper on Axiom of choice and you want to first state its definition, which definition do you use and why?

Answer 1

The whole point of "indexed family" is that it is a function. It is a function mapping the index to some object.

So if you consider the first statement, it just says in a more complicated language that every family of non-empty sets has a choice function. Why more complicated? Because it requires the understanding that an indexed family is a function, whereas the second formulation requires just knowing what is a function.

If I had to define the axiom of choice formally, I'd opt for the simplest formulation in terms of "primitive notions". In the case of set theory, this means the following:

If $S$ is a family of pairwise disjoint and non-empty sets, then there exists $C$ such that for all $X\in S$, $C\cap X$ is a singleton.

Or just the definition with a choice function. Talking about "indexed families" requires us to understand what that means in terms of sets; and one leading approach to axioms is simplicity: axioms should be simple, and require as least "interpreted structure" as possible.

Answer 2

The mostly commonly used AC has 3 forms:

1. choice function: suppose you have a bag of inhabited sets $A$, then you can just say: oh, let $f$ be a function on $A$, with the action $f(x)\in x$ for each $x\in A$.

2. zorn's lemma: suppose you have a partial order $A$ with the property that every linearly ordered subset (chain) is bounded above, then you can say: let $x\in A$ be a maximal element!

3. well-ordering principle: suppose you were given any set $A$, you can say: oh, let $<$ be a well-order on $A$!

AC is super non-intuitionistic by nature because it implies law of excluded middle, so use with caution!

The first definition you saw on wiki is not preferable because it is not intuitive how the new set is formed. That set exists simply because the AC claims it exists. It is included because you don't have to define the notion of function in order to talk about it.

regarding your last question, I think the choice function is the standard one and it probably is the easiest version to talk to a pedestrian on the street imho.