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Full Idea
The iterative conception justifies Power Set, but cannot justify a satisfactory theory of von Neumann ordinals, so ZFC appropriates Replacement from NBG set theory.
Gist of Idea
The iterative conception has to appropriate Replacement, to justify the ordinals
Source
Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9)
Book Ref
Hossack, Keith: 'Knowledge and the Philosophy of Number' [Routledge 2021], p.146
A Reaction
The modern approach to axioms, where we want to prove something so we just add an axiom that does the job.
Related Idea
Idea 23625 Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack]
| 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] |