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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.

XT Chen
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    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. – hmakholm left over Monica Aug 31 '19 at 17:04

1 Answers1

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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.

JunderscoreH
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  • 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. – XT Chen Aug 31 '19 at 16:59
  • @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). – hmakholm left over Monica Aug 31 '19 at 17:06
  • @HenningMakholm Then what is a class first? – XT Chen Aug 31 '19 at 17:07
  • @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. – hmakholm left over Monica Aug 31 '19 at 17:10
  • @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. – JunderscoreH Aug 31 '19 at 17:15
  • @JunderscoreH So the collection of all classes not containing itself is not a thing existing, i.e. not in the universe. Is my comprehension right? – XT Chen Aug 31 '19 at 17:19
  • @ExactSequence Yes. In set theory and in class theory, it doesn't exist. – JunderscoreH Aug 31 '19 at 17:23
  • @JunderscoreH In ZFC if $\phi (x,p_1,\dots , p_n)$ is a formula then $ C ={ x : \phi (x, p_1,\dots , p_n) }$ is a class. Then what is $x$ in $\phi$? Is it a thing existing, i.e. a set? So similarly an object in a formula in NBG must be a class, am I right? – XT Chen Aug 31 '19 at 17:32
  • @ExactSequence: $x$ is a placeholder for each set that might or might not be an element of $C$. The class has as elements, by definition, every set that satisfies $\phi$. If there happens to be a non-set class that satisfies $\phi$ (as there well might), it will nevertheless not be an element of $C$. – hmakholm left over Monica Aug 31 '19 at 17:35
  • @HenningMakholm Therefore in ZFC only sets are considering in formula? – XT Chen Aug 31 '19 at 17:37
  • @ExactSequence: Whoops, I missed that you wrote "in ZFC". I was describing the situation in NBG. In ZFC, writing ${x : \phi(x,p_1,\ldots,p_n)}$ might not correspond to anything at all. The axioms of ZFC do not promise you that the universe contains anything that fits that description. – hmakholm left over Monica Aug 31 '19 at 17:42
  • @ExactSequence You can use set-builder notation to define all sorts of collections, but it doesn't mean that those collections actually exist. In any theory, the variables range over the domain of discourse. In set theory, this means sets. In class theory, this means sets as well as classes. In either case, the defined collection might not exist. So in set theory, $C={x:\phi(x,\vec{p})}$ is a collection of sets, and so a class, and thus might not exist. In class theory, $D={x:\phi(x,\vec{p})}$ similarly is a collection of classes, but might not exist (e.g. if a class is an element). – JunderscoreH Aug 31 '19 at 17:43
  • @JunderscoreH Hungerford considered 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 $A$. So it is actually not a definition of class in NBG. And I come up with another question: what is belonging '$\in$'? – XT Chen Aug 31 '19 at 17:52
  • @ExactSequence Hungerford's definition would be the definition of a class from the perspective of set theory (because proper classes are objects in class theory). Hungerford phrasing it as "it is possible to determine" isn't quite right, since that alludes to computability. Basically, Hungerford is adopting set theory, and saying that a class is a collection ${x:\phi(x,\vec{p})}$ where $\phi$ is a formula (with parameters $\vec{p}$). What is membership? It's just membership: $a\in{a,b,c}$, $\pi\in\mathbb{R}$, for example. You want to ask a separate question for more depth – JunderscoreH Aug 31 '19 at 18:05
  • @JunderscoreH Ok. Thank you first because you make me more clear about this. And a new question was created. – XT Chen Aug 31 '19 at 18:18
  • https://math.stackexchange.com/questions/3340300/what-is-the-definition-of-belonging-in-axiomatic-set-theory – XT Chen Aug 31 '19 at 18:26