TITLE: Is the notion of a set always a primitive notion?
QUESTION [7 upvotes]: Is there an approach in set theory where we have a definition of a set, or do we always consider the notion of a set as being a primitive notion that cannot be defined in terms of previously defined other notions?

REPLY [2 votes]: There are candidates for foundational theories, such as category theory and the theories that Rob mentions, in which the notion of "set" is not primitive, but I don't think these count as set theories even if they turn out to be mutually interpretable with set theories.
In set theory you can get close to what you want, I think, with NBG set theory (the one-sorted formulation) in which "class" is the primitive notion and a set is defined as a class that is contained in some other class.