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I don't quite understand the difference between sets and classes.

A class consists of sets. Why is not a class a set?

A group itself is a set. A isomorphic class of group is not a set, right?

Could you explain how to distinguish a set from a class.

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@Martin: Do we really need a tag for proper classes? I think not. –  Asaf Karagila Mar 24 '13 at 9:34
    
@Asaf I wasn't sure either, but since the tag already existed, I've added it to this question, too. (If we keep the tag, this question is definitely appropriate for it.) I have no objections against removing the tag. (I don't really have preference either way, whether to keep it or to remove it. But at least this retagging made some more users aware of the existence of this tag, so it was not totally useless.) –  Martin Sleziak Mar 24 '13 at 9:48
    
@Martin, in this case I think that we should remove it altogether. –  Asaf Karagila Mar 24 '13 at 10:56

4 Answers 4

Working in the set theory ZFC, all reasoning about classes is strictly informal. A class is informally taken to consist of all sets satisfying some condition. For example, the 'universe' is the collection of sets $x$ which satisfy $x=x$. More formally, two formulae $\phi$ and $\psi$ (with one free variable) refer to the same class if $\forall x(\phi(x) \leftrightarrow \psi(x))$, and a set $x$ is a 'member' of the class referred to by $\phi$ if and only if $\phi(x)$ is true. We denote this class by $\{ x\, :\, \phi(x)\}$, so for example $\{ x\, :\, x=x \}$ is the universe, the class of all sets.

All sets are themselves classes, but it's possible for a class not to be a set. Let $V$ denote the class of all sets satisfying $x=x$. If $V$ were a set then by the axiom schema of separation would dictate that $W = \{ x \in V\, :\, x \not \in x \}$ were a set. But then $W \in W \leftrightarrow W \not \in W$, which is obviously nonsense; so the bit where we went wrong must have been to assume that $V$ is a set! This is Russell's paradox.

Some set theories treat classes as formal objects, such as NBG and MK.

Anyway, yes, a group is defined to be a set $G$ with some operation on it, but the class of groups isomorphic to $G$ cannot be a set. To see a simple example, the isomorphism class of the trivial group contains $\{ x \}$ for each set $x$, with the trivial group operation $x \cdot x = x$. The class of all such groups cannot be a set since if it were then it would biject with the universe via $\{x\} \mapsto x$, thus making the universe a set by the axiom schema of replacement (contradicting what we saw above).

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In addition to all that has been written here, and before (see links below), let me give you the most informal intuition on this subject.

Consider the natural numbers $\Bbb N$. Each proper initial segment is finite, and in fact represents a particular finite number. But there is no number represented by $\Bbb N$. Why? Because $\Bbb N$ is too large to be a natural number. Natural numbers are finite, but collections of natural numbers don't have to be finite.

In modern set theories such as ZFC, the objects of the universe are sets. And every set has a size, or cardinality. Classes are those collections of sets that do not have a size. Much like $\Bbb N$ is an initial segment which is too big to be finite, classes are collections which are too big to be sets. (One remark is that in some set theories classes are objects and have measurable sizes, but those go beyond the scope of ZFC and naive set theory.)

Some links, where many words have been written:

  1. difference between class, set , family and collection
  2. Difference between a class and a set
  3. What should we call the 'sets' which don't exist under certain set theory axioms?
  4. Does $A=\{a|\forall x\in \emptyset\ H(x,a) \}$ make sense?
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Note that Ittay's example from some comments on this page, of a singleton whose element is a proper class, $\{\{x\mid x=x\}\}$ is not a collection of sets, so the fact that "it has a size" is irrelevant. –  Asaf Karagila Mar 24 '13 at 0:41
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+1 excellent comparison with the naturals. –  Ittay Weiss Mar 24 '13 at 0:56
    
Can the downvoter explain? –  Asaf Karagila Jul 13 '13 at 15:45

Naively, every collection of things forms a set. However, Russell's paradox shows that such naive construction lead to contradiction. Thus, there is a need to be more precise. One way of doing it is to say: we can't guarantee no contradictions for all conceivable collections of things, but perhaps we can guarantee that for some smaller collection of collections there are no problems. So, we set out to make a distinction between collections that are ok, and we call them sets, and collections that may not be ok, which we call proper classes. We can't actually prove that for sets there are no problems but we hope it's the case and work under the assumption that a model of sets exists.

Within this model, as long as the entities you are concerned with are sets then you can do all of the standard things we do with sets, and no such construction will lead to a contradiction. If however you deal with some proper classes then the model says "you are on you own here buddy". The model can't guarantee that anything you do with proper classes will be safe. It might, but who knows.

More formally, and to answer your questions, the elements in a class are all sets. But not all classes are themselves sets. A group does have an underlying set. An isomorphism class of a group though is not a set but is a proper class. I hope this helps.

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Notice all this confusion goes away if one abandons ZFC for some "equivalent" system that admits the existence of proper classes.It made sense to reject these other systems for the simplicity of ZFC before World War II,but I think the existence of category theory and it's constructions have changed this dramatically. At least, that's how I see it from a non-expert's perspective. –  Mathemagician1234 Apr 27 at 17:25

Informally, a class is any collection of objects, potentially one that is "too large" to be called a set. For example, the class of all sets is a class that is not a set, because if we called it a set, then we would introduce paradoxes like Russell's paradox into set theory. Sets are classes, but not all classes are sets.

More formally, in the first-order language of set theory (in ZFC for concreteness), classes are another way of talking about first-order formulas in the metalanguage. That is, if $\phi$ is a first-order formula in the language of set theory, then we get a "class" $$A_\phi=\{x\mid\phi(x)\}.$$ We say that the class $A_\phi$ "is a set" if there exists a set $y$ such that $$\forall x(x\in y \leftrightarrow \phi(x));$$in this case we would write (abusing notation) $y=A$, but this would really just be shorthand for the formula above. Similarly we define expressions like "$y\in A_\phi$" to mean just the formula $\phi(y)$. Thus, working with classes is the same as working with first-order formulas. Moreover, as I mentioned above, every set "is a class." Indeed, if we have a set $x$, then we can define the class $$A=\{y\mid y\in x\}.$$ Then $A=x$ in the sense defined above. The reason that not all classes are sets is that not all first-order formulas in the language of set theory give rise to sets. For example, if $\phi$ is the formula $x=x$, then the class $$V=\{x\mid \phi(x)\}$$ would contain every set. If $V$ were a set, then we would get problems like Russell's paradox.

As Ittay Weiss notes, classes are not actually present in ZFC; they are only used in the metalanguage.

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It should be noted that in ZFC there are no classes, so any mention of classes as you describe above for ZFC is in the ambient metalanguage. There are formalisms that allow for classes in the model (e.g., Goedel-Bernays) –  Ittay Weiss Mar 23 '13 at 23:50
    
and it's not only that classes are like sets that are simply too big. $\{\{x\mid x=x\}\}$ is a proper class containing just one element, so it's quite small. –  Ittay Weiss Mar 23 '13 at 23:53
    
@Ittay Weiss Thank you for your first comment; I realize now that I wasn't being very clear about whether or not classes are actually objects in ZFC. Regarding your second comment, how would the proper class $\{\{x\mid x=x\}\}$ contain anything at all? Anything that is an element of $\{\{x\mid x=x\}\}$ would have to be a set, but $\{\{x\mid x=x\}\}$ contains no sets (I don't want to say that a proper class is an element of something else). Am I misunderstanding something? –  Artem Mavrin Mar 24 '13 at 0:03
    
well, is the thing I describe just the empty set? doesn't look empty to me. The point is, I think, that without some more precise axioms for set theory we can discuss classes endlessly, but can't really argue about them. –  Ittay Weiss Mar 24 '13 at 0:07
    
@Ittay: If one defines a proper class as a collection of sets which is not a set, then $\{\{x\mid x=x\}\}$ is neither a proper class nor a set. It's a collection in the meta-meta-universe (soon Abed Nadir will pop by for a meta-reference). See my answer, and the comment below it. –  Asaf Karagila Mar 24 '13 at 0:47

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