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Full Idea
If a particular set-theory is in a strong sense 'reducible to' the theory of ordinal numbers... then we can still ask, but which is really which?
Gist of Idea
If ordinal numbers are 'reducible to' some set-theory, then which is which?
Source
Paul Benacerraf (What Numbers Could Not Be [1965], IIIB)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.290
A Reaction
A nice question about all reductions. If we reduce mind to brain, does that mean that brain is really just mind. To have a direction (up/down?), reduction must lead to explanation in a single direction only. Do numbers explain sets?
Related Idea
Idea 10687 Maybe we reduce sets to ordinals, rather than the other way round [Hossack]
| 16896 | If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge] |
| 18161 | The theory of classes is superfluous in mathematics [Wittgenstein] |
| 8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend] |
| 8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe] |
| 9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf] |
| 17821 | You can ask all sorts of numerical questions about any one given set [Yourgrau] |
| 17815 | We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau] |
| 10560 | Set-theoretic imperialists think sets can represent every mathematical object [Fine,K] |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
| 9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
| 9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |