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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]
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
| 23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
| 23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
| 23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
| 23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |