Full Idea
That the iterative sets suffice for most of ZFC does not show they are necessary, nor is it evident that the set of operations has no fixed points (as 0 is a fixed point for square-of), and no infinitely descending chains (like negative integers).
Clarification
ZFC is the Zermelo-Fraenkel-with Choice system
Gist of Idea
The iterative conception may not be necessary, and may have fixed points or infinitely descending chains
Source
William D. Hart (The Evolution of Logic [2010], 3)
Book Reference
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.80
A Reaction
People don't seem to worry that they aren't 'necessary', and further measures are possible to block infinitely descending chains.