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?