TITLE: If an identity function is its own inverse then why are its domain and co-domain the same at the same time?
QUESTION [0 upvotes]: Just to clarify, I mean that if an identity function is its own inverse, then doesn't that suggest its domain and co-domain (the same set) are different at different times, even if they're the same set. What is the rule which says that if I map the natural numbers to the natural numbers via the identity function, that the domain and co-domain are the same? Can't I say that I'm instantiating two different versions of the same set, and that I'm mapping id(x) in the domain to element x in the co-domain? Then when I invert the function, the first and second version of the set switch around? 
I think I understand the mathematics behind identity and invertible functions, but maybe this is something more fundamental and philosophical in set theory which I'm missing.

REPLY [1 votes]: Although I do not know much about the many set-theories that exist, the generally accepted one (Zermelo–Fraenkel set theory) is axiomatic, and what constitutes a set is not decided by a definition but by axioms. In particular, there is an axiom stating that a set is identified by its elements. So two instances of the same set (as you put it), are in-fact the same set.
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