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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 Ref
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.
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos] |
| 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
| 17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
| 10565 | There is no stage at which we can take all the sets to have been generated [Fine,K] |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| 9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
| 10405 | In the iterative conception of sets, they form a natural hierarchy [Swoyer] |
| 10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |