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Full Idea
Any set has k members if and only if it can be put into one-to-one correspondence with the set of numbers less than or equal to k.
Gist of Idea
A set has k members if it one-one corresponds with the numbers less than or equal to k
Source
Paul Benacerraf (What Numbers Could Not Be [1965], I)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.275
A Reaction
This is 'Ernie's' view of things in the paper. This defines the finite cardinal numbers in terms of the finite ordinal numbers. He has already said that the set of numbers is well-ordered.
| 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] |