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I don't understand the difference between a set and a class. The definition which I studied is:
A set $A$ is a class such that there exists a class $B$ such that $A \in B$.
But isn't it always true as we can have a collection of all sub classes of $A$, which will again be a class. Something like power set, but in this case I should call it a power class. Somebody please clarify.

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    $\begingroup$ give a link of the reference please $\endgroup$ – user 1 Jan 11 '15 at 9:05
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    $\begingroup$ Have you read the many questions of people with the same confusion, and the many answers given to those questions? $\endgroup$ – Asaf Karagila Jan 11 '15 at 9:06
  • $\begingroup$ But my confusion is why don't a power class work in the condition given for a set? $\endgroup$ – akansha Jan 11 '15 at 9:18
  • $\begingroup$ @akansha What axiom gives you a powerset for a class? $\endgroup$ – Henno Brandsma Jan 11 '15 at 9:48
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    $\begingroup$ @akansha You are always working under some set of axioms, and you can construct new ones from old ones using those axioms. There is no axiom that says a power class for a class exists. Just one for sets. And adding such an axiom gives paradoxes (what is the power class of the class of all sets?). $\endgroup$ – Henno Brandsma Jan 11 '15 at 10:21
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Set theory is a mathematical theory, like any other mathematical theory it has a "universe", and the axioms and properties are required to hold within the universe.

In set theory, the objects in the universe are called sets.

But since we don't live inside that universe, but rather work from the outside (in one way or another) we are free to talk about collections of elements from that universe. We can talk about "all the rational numbers which are negative or their square is strictly smaller than $2$", and we can talk about the collection of all sets which have a certain property.

What is confusing in the case of set theory is that sets come to model, mathematically, the notion of a collection of mathematical objects. So if classes are also collections are mathematical objects, why aren't classes sets?

As it turns out, not every collection which we can define form a set. That's the essence of Russell's paradox.

So we limit ourselves to some collections of mathematical objects, and require that they satisfy certain axioms. Classes are collections which need not be in our universe, and therefore don't have to satisfy the axioms of set theory.

Let me reiterate this point. Sets are elements of the model of set theory, and they have to satisfy the axioms, e.g. the axiom of power set. Classes are collections of elements from a model of set theory, but they don't have to correspond to any element in the model, and they don't have to obey the axioms. Just like a real number can be seen as a set of rational numbers, but it doesn't mean that it can be written as a quotient of two integers.

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  • $\begingroup$ He might be working in NBG set theory? $\endgroup$ – Henno Brandsma Jan 11 '15 at 10:41
  • $\begingroup$ Maybe he's working (she's working?) in type theory? The post didn't specify, so I assumed it was $\sf ZFC$ or some variant thereof. $\endgroup$ – Asaf Karagila Jan 11 '15 at 10:48
  • $\begingroup$ In NBG theory the definition of a set which is a class which is a member of another class. In ZFC (how do you get the nice font??) I'm not aware of class being defined in that way (it's more a metamathematical notion). Hence my idea of NBG; I'm not familiar with type theory. $\endgroup$ – Henno Brandsma Jan 11 '15 at 10:52
  • $\begingroup$ In $\sf ZFC$ (\mathsf or \sf) we define classes informally, or in the meta theory, as collections definable (with parameters) over the universe. Sets happen to be definable trivially with themselves as parameters, and this means that a class is a set if it is an element of another set. $\endgroup$ – Asaf Karagila Jan 11 '15 at 10:56
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Sets are strongly regulated by some axioms and only those objects that satisfies these axioms are called sets.

Classes are also used in mathematics. For example, the class of all sets includes any object having the property of being a set. Classes are used e.g. in category theory to make a mathematical object of a property.

But it's a good thing in general to be able to strictly define a set.

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