Thre are many sets with a single object, for example the set which only contains the statue of liberty or the set which contains my copy of Catch-22.
But how many sets are there that contain nothing?
On the face of it there seems to be just the one. The empty set is unique in its particularity as it contains nothing to distinguish itself.
On the other hand one could say that there are many empty sets, and that they are all identical - as in isomorphic - but not identical - as in they are exactly the same. To make this more concrete, consider Jamila & Jemima each carrying a basket, and in Jamilas basket there is an emerald, and in Jemimas basket there is a ruby. So the contents of their baskets are not identical. If they empty out their baskets. The contents of the basket are now identical (isomorphic) but the two baskets are not identical.
Which is the more accurate view?
Is it correct to say in formalised ZFC that there is just one empty set?
Is it correct to say that in the categorical set theory, say ETCS, that there are many empty sets but they are all isomorphic?
I am less interested in the 'formal' parts of the question as opposed to the conceptual arguments about uniqueness or non-uniqueness of empty 'sets' - where sets shouldn't be thought of as in ZFC
It does turn out that in ETCS there are more than one empty set (and since they are intial) they are also all isomorphic to each other. See the anser to this question