The class of all classes not containing themselves

set-theory paradoxes

Solution 1:

Classes in ZF are merely collections defined by a formula, that is $A=\{x\mid \varphi(x)\}$ for some formula $\varphi$.

It is obvious from this that every set is a class. However proper classes are not sets (as that would induce paradoxes). This means, in turn, that classes are not elements of other classes.

Thus discussion on "the classes of all classes that do not contain themselves" is essentially talking about sets again, which we already resolved.

Of course if you allow classes, and allow classes of classes (also known as hyper-classes or 2-classes) then the same logic applies you have have another level of a collection which you can define but is not an object of your universe.

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