When we want to apply Axiom of Choice , is it necessary for us to check the class we construct is a set rather than a proper class?

Question

In set theory , if we want to use axiom of choice , we have to check the class we construct is a set first . However , in some other courses (functional analysis or topology) , the textbook usually apply axiom (especially Zorn's lemma) without verify the class they construct is actually a set . So when we want to apply axiom of choice in other courses in mathematics , is it necessary for us to check the class we construct first ? Or we assume all the class we construct are the set (which is not the cases in set theory) ?

Answer 1

You are not giving a concrete example, so it's hard to tell exactly. But yes, you need to make sure that you're dealing with a set. You can define a version of AC that works with proper classes, but that is not provable in ZFC that this version holds (it is consistent, though).

In most textbooks, the constructions are usually things like an ideal in a ring, algebraic extension, or a particular sequence/net/topology, etc. and in those cases you are always bound within a set.

Each case is separate, so there is no "global way" of answering your question. But the only case which is worth mentioning is something like field extensions, where there is a proper class of algebraic field extensions for any given field (unless it is algebraically closed), simply because any object can be added as a root of a polynomial. However, we can prove that up to isomorphism, there is a set of representatives. So that means we can fix a large enough set and assume all the relevant fields come from that set.

The same holds in other situations where classes might be involved.

(Caveat lector: Yes, it is possible to talk about proper classes such as the surreal numbers, or the class of ordinals, and somehow try and apply Zorn's lemma in those situations. And yes, we do need to be more careful when we apply it there if we are in the confines of ZFC. But since ZFC is not suited for talking about proper classes all that much anyway, the standard solution is to move to Gödel–Bernays set theory, or even Kelley–Morse, and there the standard is to take global choice as an axiom.)