I am trying to understand this proof of Zorn's lemma.
I think the notation is clear to me and I can follow the proof until the point where the author defines conforming subsets although I am not entirely sure if I get the part using the axiom of choice.
Now my questions are:
1) Why do we need the axiom of choice? Until now I have viewed the axiom of choice as "we can create a new set from other sets even if we have uncountably many sets". Now wikipedia states it somewhat differently, i.e. that the axiom of choice gurantess a choice function, but I can see this formalizes my intuition since if such a function exists then we can use this function to pick elements from each set in a collection of sets. However, I am not entirely sure why the argument in the proof of Zorn's lemma requires a choice function. Given the assumptions we know that every chain in $X$ has a strict upper bound, so what's the use of the axiom of choice now?
2) Why does the author define the conforming property for arbitrary subsets? In the definition he uses an initial segment $P(A,x)$, but in the notation part he defined an initial segment only for chains. It also makes sense to define it only for chains since if a subset does not have a total order, then some elements might simpy not be in $P(A,x)$ because there is no relation between $x$ and those other elements.
I am quiet new to such deep set theory arguments and only want to understand this since it is relevant for a lemma on extensions of solutions to ODEs.
Thanks for any help and suggestions!