Franklin writes: << Regarding defining a specific function, I am thinking in terms of ZF set theory. There, a function is just a set of ordered pairs (of sets) with the requisite uniqueness property. Such a function does not bring with it any concept of its codomain, just the image - which is normally called the range. The only general range would be the set of all sets - which does not exist.

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I would be interested to learn on what basis you state that ZF has a definition of function that agrees with what you write above. To the best of my knowledge the only kind of object in ZF is a set; the word "function" does not play a role. I am also not clear about what question your last sentence about "general range" is attempting to answer. --Dan