In general, the term limit is defined topologically: any topological space comes with its own natural definition of limit. But it's not at all clear to me whether "general set convergence" is a limit in this particular sense. Is there a topology on Set so that the "natural" definition of limit coincides with general set convergence?
If not, then I guess K_h has a point about the naturalness of this definition of limit (and, conversely, if so, then K_h has no good point at all).
-- "Eventually the truth will come out, and you know what I'll do then? Probably go to the beach. I'll also hang out in some bars. Yup, I'll definitely hang out in some bars, preferably near a beach." -- JSH on the rewards of winning a mathematical revolution