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Full Idea
The transfinite ordinal numbers are important in the theory of proofs, and essential in the theory of recursive functions and computability. Mathematics would be incomplete without them.
Gist of Idea
Transfinite ordinals are needed in proof theory, and for recursive functions and computability
Source
Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.1)
Book Ref
Hossack, Keith: 'Knowledge and the Philosophy of Number' [Routledge 2021], p.152
A Reaction
Hossack offers this as proof that the numbers are not human conceptual creations, but must exist beyond the range of our intellects. Hm.
Related Idea
Idea 23622 We can only mentally construct potential infinities, but maths needs actual infinities [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] |