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".

The question, however, is what is this all good for? What can you achieve in this manner that you couldn't achieve with just talking about sets? Yes, you gain access to more collections. But you can just strengthen your set theory and get access to more collections and use that as a simpler mechanism.

  • $\begingroup$ Asaf Karagila:I saw that some people have asked the similar questions and you involved in.Based on your answer,I have two questions and I hope you can answer for me.Thanks a lot!1.Acturally I can't quite understand what you want to clarify in your last paragraph.Do you mean my point of view is correct but in terms of the simplicity we only want to deal with sets.Therefore those big collection is "meaningless" so we don't want to deal with that?2.Is my question run into type theory rather than set theory?If it is,can you recommand some book that related to this? $\endgroup$ – majin vegeta Jul 29 '16 at 6:54
  • $\begingroup$ :Furthermore,as what I was mentioned above:the problem is due to the improper language.Therefore I think this maybe my question is included in mathematical logic.Is this make sense? $\endgroup$ – majin vegeta Jul 29 '16 at 6:57
  • $\begingroup$ 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. $\endgroup$ – Asaf Karagila Jul 29 '16 at 7:19
  • $\begingroup$ :It is hard to describe what the exact image of "α-class" is..However,my crucial question is whether or not the "collection" is unbounded,In other words,given a "collection"we also can construct a higher level of "collection" theortically.Such as X is a "collection",than we may define P(X) is all the combination of "stuff".It is obvious that P(X) has more "stuff" than X.In order to avoid the paradox,I use the word "collection" and "stuff" to replace to word "class" and "element". $\endgroup$ – majin vegeta Jul 29 '16 at 15:41

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