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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
up vote 3 down vote accepted

The distinction is meaningful in many senses. Certainly it is on ZF foundations, and contrary to what Dylan Wilson might seem to suggest, the distinction can be stated formally in the Elementary Theory of the Category of Sets as well as in the standard axioms for the Category of Categories as Foundation.

I do not see any reason to use any term other than "isomorphism class." That is a vague question.

As to the third question, there is no important difference. You know you could require just one terminal set. In ETCS or CCAF you could add a skeletal axiom saying there is just one set in each isomorphism class. And on ZF foundations even without changing the definition of function you could pick one singleton and insist it is the only singleton allowed in your category. It would not achieve anything important -- as you can sort of see from the triviality yet artificiality of the ZF way of doing it.

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