# Is there exists unbounded collections rather than be restricted into the world “class”?

In set theory,there is a question states that whether the class which contains all the classes exists? The answer is "NO" obviously.Otherwise it will lead to paradox.The reason is that there is a problem with the predicate:"class"which contains all the classes.

Like the classical paradox--the set of all the sets which cannot exist.So to be instead,we use the world--"universal class U" to replace the word "set" to define a collection which contains all the sets in order to avoid the logical paradox.

Similarly,I can use another word to define a collection which contains all the classes(some people say this is 2-class,let me admit this word here,so 2-class is not a class,it is sufficient to avoid the logical paradox). Phylosophically,I think this collection can be defined.As the "2-class" contains all the subclasses of "U".Indeed,"2-class" contains all the possible combination of the elements in"U" whereas "U" only have the singeletons.So it is obvious that "2-class"contains "more" than "U"Similarly we can define "α-class"

So phylosophically,does anyone think this is plausible,or someone think there do have a collection which there is no more collection contains more thing than that collection?

No, the solution in $\sf ZFC$ and in all its class-extensions ($\sf NBG,KM$, etc.) is that classes which are elements of other classes are sets. So there is no class of all classes, because no proper class is an element of another.
You can extend class-theories to also include 2-classes and so on. There is nothing fundamentally wrong with that. But pretty quickly you run into a simple type theory situation. Because now you have sets which we might as well call $0$-classes, then $1$-classes and $n$-classes and so on. And each of those is a different kind of object. And you will probably want to add new axioms like the existence of power set-like $n+1$-classes or comprehension axioms for $n$-classes, for every $n$, just to ensure that these collections behave "reasonably".
• No, my point is that you can suggest a lot of things when it comes to "something is amiss from such and such". The real hard work is to find something missing that is actually useful, mathematically speaking of course, rather than just playing with changing variables of a formula. Sure, we can have $n$-classes, but you probably want to include some axioms that govern them. At what point do you just return to having sets with a hierarchy (like you already did in ZF)? As for type theory, I'm not the person to help you there, I don't know much about it. – Asaf Karagila Jul 29 '16 at 7:19