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Full Idea
Benacerraf claims that the concept of a progression is in some way the fundamental arithmetical notion, essential to understanding the idea of a finite cardinal, with a grasp of progressions sufficing for grasping finite cardinals.
Gist of Idea
To understand finite cardinals, it is necessary and sufficient to understand progressions
Source
report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xv
Book Ref
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.117
A Reaction
He cites Dedekind (and hence the Peano Axioms) as the source of this. The interest is that progression seems to be fundamental to ordianls, but this claims it is also fundamental to cardinals. Note that in the first instance they are finite.
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 14146 | We aren't sure if one cardinal number is always bigger than another [Russell] |
| 13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C] |
| 17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf] |
| 17906 | To explain numbers you must also explain cardinality, the counting of things [Benacerraf] |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| 17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |