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As I understand it, in standard axiomatic set/class theories, a proper class is a class that isn’t a member of anything else. Now, we often do model theory with such theories, yielding for example the result that NBG is a conservative extension of ZFC. But in order to do that, we must be able to talk about a model of the theory. How can any collection (either a set or proper class) be a model of a theory like NBG if no such collection can exist (it can’t house the proper classes?)

(It also doesn't seem to me like it would matter if we declared that this collection was a type beyond "set" or "class", since ultimately we'd still run into the same problem with that type too.)

Nihal Uppugunduri
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    Models are sets with extra structure. A model of a class theory like NBG consists of two sets: the elements of the first set are called "sets" and the elements of the second set are called "classes". The extra structure is a binary relation between the elements of these sets called "$\in$". This $\in$ relation does not necessarily have anything to do with the real membership relation between real sets. – Alex Kruckman Dec 02 '23 at 02:05
  • @AlexKruckman Ah OK, so a model of NBG would just be any set or class that satisfies the axioms, not necessarily the "collection of all classes." But then would that mean that model-theoretic results about NBG don't apply to "all" classes? Or we could probably take a "universal working set" U and then take the set of subsets of U to be a model of some axiomatic set theory, and then say that since U is arbitrary our model-theoretic results apply to all sets. Can we then dispense with the notion of class entirely, restricting ourselves to subsets of U and yielding equivalent results? – Nihal Uppugunduri Dec 02 '23 at 03:28
  • Not "set or class that satisfies the axioms", set that satisfies the axioms. See Carl Mummert's answer to the linked duplicate. – Alex Kruckman Dec 02 '23 at 03:38
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    You should distinguish between proving things in ZFC/NBG from proving things about ZFC/NBG. Model-theoretic methods are useful for the latter, not so much the former. E.g. if you want to prove a theorem in NBG, you just write down a proof. If you want to prove "NBG does not prove $P$" (this is a meta-theoretic statement statement about NBG), you provide a model of NBG in which $P$ is false. – Alex Kruckman Dec 02 '23 at 03:44
  • @AlexKruckman Ah OK, thanks I think that clarifies it for me. So would it be correct to say: model-theoretic results about NBG apply to any sets that satisfy the axioms, while if we want to prove something about classes, then we just work in NBG and write proofs that follow the axioms of NBG, without ever referring to all classes as a model of NBG? – Nihal Uppugunduri Dec 02 '23 at 23:20
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    You can think of the axioms of NBG as facts about the "real" universe of sets and classes. How do we learn facts about the "real" universe? We start with some basic facts that we can agree on, and proceed to prove logical consequences of those facts. OTOH, you can think about models as little toy universes that happen to satisfy all our basic axioms (but are small enough to be sets). By Gödel's Completeness theorem, a sentence is provable from our axioms if and only if it's true in all models, but there is no need to think about models when proving theorems about the "real" universe. – Alex Kruckman Dec 04 '23 at 16:13

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This is an answer to my own question, based on Alex Kruckman's comments. My question was related to another Math StackExchange question ("How can there be genuine models of set theory?"), but I think it was different enough to not be marked as a duplicate.

We can view an axiomatic set/class theory like NBG in two ways. One is as rules for validly manipulating and working with classes. Then, in order to use classes, we just need to make sure we follow the rules of the theory. Another is as axioms for any collection of objects, together with a "membership" relation between them, that satisfies the axioms. These objects need not be sets or classes (in fact such a collection can't have proper classes as elements), and the membership need not correspond to "traditional" set membership. This is just like how a model of $a+b=b+a$ need not be the set of integers, and $+$ need not be addition of numbers. The model theory of a theory like NBG applies to the second usage, not the first.

So there is no contradiction in terms of validity. We can work within NBG and follow its rules to use classes correctly, or we can study the models of NBG just as we study models of the group axioms or ring axioms. Furthermore, models are always assumed to be sets, so this distinction would apply to any axiomatic set theory without classes too, like ZFC (since we can't have the set of all sets.) (We can study proper class-sized models too, but they are then called "class models" or "inner models." The term "model" without qualification is assumed to be a set.)

Another way to look at this is as follows. Sometimes when working say with vector spaces, we call the elements of an arbitary vector space "vectors" even if they're not elements of $\mathbb{R}^n$, which are the "standard" vectors. Similarly, when we have some model of an axiomatic set theory like ZFC or NBG, we may call its elements "sets" in that context even though they are not actually sets in the standard sense. We call a standard set an external set, and we call an element of a model of ZFC/NBG/etc. an internal set. Then, the distinction comes down to external vs internal sets.

Nihal Uppugunduri
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