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For a collection $C$ define an object $b \in C$ iff $b$ is a class and $b \notin b$ and otherwise $b \in C$. Then we can decide any object whether it belongs to $C$. In Hungerford's Algebra p.2 it states in illustrating Gödel-Bernays axiomatic set theory that

Intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or element) of $A$

Since it's possible to determine whether an object belongs to $C$, $C$ must be a class.

Then there is a paradox whether $C \in C$. How to avoid this in Gödel-Bernays axiomatic set theory.

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marked as duplicate by Matthew Daly, The Count, nmasanta, Leucippus, José Carlos Santos Sep 1 at 9:09

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    $\begingroup$ I don't think this is a duplicate, since it explicitly asks about the relation to NBG, and an answer that explains that would not really be on point for the earlier question. $\endgroup$ – Henning Makholm Aug 31 at 17:04
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In axiomatic class theory like Von Neumann–Gödel–Bernays, $\mathsf{NGB}$, only sets can be elements, so no proper class belongs to anything, much less itself. In a typical set theory, classes do not exist as objects that can be collected together but are instead meta-theoretic. Class theories kind of continue this perspective by limiting what we can consider as collections beyond sets. So $C$, as you defined it, wouldn't exist.

The issue with allowing us to consider any collection to exist is precisely your worry: Russell's paradox. So set and class theories like $\mathsf{ZFC}$ or $\mathsf{NGB}$ give a kind of iterative concept of collection. If we consider the collection of all sets, we don't get a set, but instead a class. And we can continue this iterative conception: if we consider the collection of all classes, we don't get a class but a 2-class. The collection of all 2-classes would be a 3-class, and so on.

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  • $\begingroup$ If only sets can be elements then what is set first? Is in axiomatic system it defines a universe and all objects in it is a set? And in Hungerford's book it says intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or element) of $A$, then it seems to be a problem in this statement. $\endgroup$ – XT Chen Aug 31 at 16:59
  • $\begingroup$ @ExactSequence: In NBG, a "set" is by definition a "class" (that is, one of the things the theory is speaking about) that happens to be an element of at least one other class. (At least that's how it works in Mendelson's influential development; I think there are other developments where sets are a separate sort). $\endgroup$ – Henning Makholm Aug 31 at 17:06
  • $\begingroup$ @HenningMakholm Then what is a class first? $\endgroup$ – XT Chen Aug 31 at 17:07
  • $\begingroup$ @ExactSequence: A "class" is the fundamental kind of object the theory is speaking about. That is, everything that can be the value of a variable in the theory's formulas is a class. $\endgroup$ – Henning Makholm Aug 31 at 17:10
  • $\begingroup$ @ExactSequence Hungerford might be mixing things up a bit for the sake of simplicity. $\mathsf{NGB}$ is a class theory: everything that exists is a class. In set theory everything that exists is a set. But often in set theory, we want to consider other kinds of collections (of things that exist). These are called classes. So note that proper classes contain sets (i.e. things that exist) but do not themselves exist in set theory (since they are not sets). Class theories like $\mathsf{NGB}$ make the same kind of distinction by defining sets as classes which belong to another class. $\endgroup$ – JunderscoreH Aug 31 at 17:15

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