In this earlier post, I asked why the axiom of choice, which is an axiom in set theory, is used in areas that are not set theory, such as group theory. The answer was that, whenever choice is used, it is used to prove something about the set of models of group theory, and hence is a statement about sets rather than about groups.
Now I’m trying to understand this for Zorn’s lemma specifically. Zorn’s lemma says:
Zorn’s Lemma — Suppose a partially ordered set $P$ has the property that every chain in $P$ has an upper bound in $P$. Then the set $P$ contains at least one maximal element.
The axiom of choice is used in the proof of Zorn’s lemma. This at first sight confused me, since Zorn’s lemma seems like a theorem about partial orders, not about sets in general. But then I realized that it is a second-order statement, rather than first order, since it is about subsets of $P$. I don’t fully understand the proof of Zorn’s lemma, since I’m somewhat confused still by the axiom of choice, so let me ask a question to clarify:
Question 1. Is it true that the proof of Zorn’s lemma uses the axiom of choice on totally ordered subsets of $P$, rather than on $P$ itself?
This gives me the more general conjecture:
Question 2. Is it true that whenever we have a first-order statement about a structure (such as posets), we never need the axiom of choice to prove it, but if the statement is higher-order, we may need it?
Note that (in the context of partial orders) I’m talking about first-order statements within the theory of partial orders, not e.g. first-order statements about the set of models of the theory of partial orders, or e.g. about the category of posets.