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Single Idea 18106

[from 'Philosophy of Mathematics' by David Bostock, in 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers ]

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 Reference

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]