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Full Idea
We do not know that of any two different cardinal numbers one must be the greater.
Gist of Idea
We aren't sure if one cardinal number is always bigger than another
Source
Bertrand Russell (The Principles of Mathematics [1903], §300)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.323
A Reaction
This was 1903, and I don't know whether the situation has changed. I find this thought extremely mind-boggling, given that cardinals are supposed to answer the question 'how many?' Presumably they can't be identical either. See Burali-Forti.
| 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] |