Is there a set theory that contains urelements, sets and proper classes all consistently?
If the answer is yes, can this set theory have proper classes as elements of urelements?
Set theory that contains urelements, sets and proper classes.
set-theory
Best Answer
The answer to the former is of course yes. Taking a variant of $\sf ZF$ which allows urelemnets, we can definable classes to have a model of a variant of $\sf NBG$ which allows urelements.
The answer to the latter question is not really. Urelements are exactly objects which have no elements but are not the empty set either; and proper classes are supposed to be collections of objects which are not in the domain of the membership relation (just in its range, when you are dealing with a theory that allows classes).
Sure, you can define something like that, but it is unhelpful, complicated, and requires you to bend over backwards just to get things running. And then it will still be equivalent to the same thing without classes inside urelements.
What people don't get about proper classes, in the standard $\sf ZF$ context, is that proper classes are collections which are too big to be an object, or a set. When we move to a class-enabled theory, then it is easy to define sets as classes which are elements of other sets. Because those theories extend what we already understand about sets and classes: the universe is made of sets, and some collections of sets are too big, and are called classes.
The thing to remember here is that syntactically, the membership relation is just a binary relation. We call them "sets" and "classes" because they satisfy properties we decided more-or-less collectively sets and classes should have. Yes, you can modify everything to get anything you want. The question is, are you still justified to call them "sets" and "classes" at the end?