# What is the difference between an empty class (which is not an element) and an empty set?

By using contradictions, in set theory we prove that an empty class

$$C = \{ x : x \text{ does not belong to } x \}$$ is not an element. By using the Axiom of Infinity and the empty set $$\emptyset$$, we can build the set of natural numbers. An empty class contains no elements and an empty set also contains no elements. How are they different?

• The first sentence makes no sense to me. What do you mean? In most set/class theories the empty set/class is unique. The $C$ in your post is not the usual definition of empty set/class. – user251257 Mar 9 at 10:30
• In fact $C$ is the proper class of all sets in NBG set theory. – user251257 Mar 9 at 10:47
• What makes you think $C$ is empty? Do you believe that every set is an element of itself? – bof Mar 9 at 11:27
• The empty class is classier. – William Elliot Mar 10 at 2:33