# Are there any consequences of the initial object in Set being unique, while the isomorphism class of terminal objects is nontrivial?

Is this even a meaningful distinction?

Is there a more appropriate term than "isomorphism class"?

Is there something different about a theory of sets with a nontrivial set-isomorphism class of empty sets?

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this is kind of an "evil" distinction; i.e. any theorem that can be stated inside category theory about the category of sets will not be able to see this. –  Dylan Wilson Aug 15 '13 at 7:15
While I'm inclined to agree with the previous comments, I do admit to having occasionally pondered the fact that the standard material set theories deliver a unique initial object but a proper class of terminal objects. –  Adam Epstein Aug 15 '13 at 7:53
That's just the axiom of extensionality, though. One also observes that the opposite of the category of complete atomic boolean algebras is equivalent to $\mathbf{Set}$ but has a proper class of initial (and terminal) objects. –  Zhen Lin Aug 15 '13 at 8:59
This is a reasonable question, but I think the answer is just "no". –  Mike Shulman Aug 15 '13 at 23:50
Perhaps a more meaningful question would be: Is there a natural category satisfying the ETCS axioms in which (a) the initial object is not unique, (b) the terminal object is unique. Of course it is very easy to construct examples, but most of them seem to be artificial. Zhen has suggested to look at the dual of the category of atomic boolean algebras. But I'm not sure if this is a ETCS category "on the nose", without using the equivalence to the category of sets. –  Martin Brandenburg Aug 16 '13 at 17:18