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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?

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    $\begingroup$ 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. $\endgroup$ – user251257 Mar 9 at 10:30
  • $\begingroup$ In fact $C$ is the proper class of all sets in NBG set theory. $\endgroup$ – user251257 Mar 9 at 10:47
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    $\begingroup$ What makes you think $C$ is empty? Do you believe that every set is an element of itself? $\endgroup$ – bof Mar 9 at 11:27
  • $\begingroup$ The empty class is classier. $\endgroup$ – William Elliot Mar 10 at 2:33

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