Why isn't there a class of all classes, in NBG set theory?

sed -i 's/set/class' russel's_paradox.txt

>>8559981
Yeah I know a class of all classes would lead to Russel's paradox complications. Hence this question.

I'm not asking a deep question about NBG or set theory in general or anything, I'm just asking a clarification question about NBG specifically.

>>8559970
OP here, Wikipedia says
>NBG does not admit "the class of all classes" (which fails because proper classes are not "objects" that can be put into classes in NBG)
and I have no idea what this means, pls help

>>8560002
By DEFINITION a proper class is something that is not contained in any other class.
So by DEFINITION if you had something that contains proper classes, it is not a class.
You can call it something else if you like, I think I've heard the term "conglomerate" being used.

>>8560002
Read Pg. 10 of Topoi: Th Categorical Analysis of Logic. NGB axiomatizes in a very systematic way the difference between sets and classes. If I remember correctly everything in NGB is a class and those objects that are members of classes are sets, or something like that. Proper classes are treated different, and the class of all classes would be a proper class.

>>8560016
Thanks, sorry I'm new to NBG (only looked it up in the first place because of category theory)

>>8560016
But what is the conglomerate of all conglomerates? A proper conglomerate? :^)

>>8560027
A category :^^)