Is there more than one instance of the Empty Set?

Question

It seems any additional instance would be equivalent in every respect to the first, hence indistinguishable, and arguably identical.

I.e., there is only one Empty Set. Correct?

Answer 1

In the most popular version of axiomatic set theory, ZF, there is an axiom, the Axiom of Extensionality, which says if two sets have the same elements they are equal.

However, there are non-extensional set theories. These are much more a minority taste. You will find some information under urelemente.

Answer 2

A set is defined by the elements it contains. The empty set contains no elements. A putative second empty set would contain the same elements, namely none, hence is equal to the empty set.