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Full Idea
The cardinal aleph-1 is identified with the first ordinal to have more than aleph-0 members, and so on.
Gist of Idea
Aleph-1 is the first ordinal that exceeds aleph-0
Source
David Bostock (Philosophy of Mathematics [2009], 5.4)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.149
A Reaction
That is, the succeeding infinite ordinals all have the same cardinal number of members (aleph-0), until the new total is triggered (at the number of the reals). This is Continuum Hypothesis territory.
Related Idea
Idea 18101 Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
| 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] |