Full Idea
If you add to the ordinals you produce many different ordinals, each measuring the length of the sequence of ordinals less than it. They each have cardinality aleph-0. The cardinality eventually increases, but we can't say where this break comes.
Gist of Idea
Each addition changes the ordinality but not the cardinality, prior to aleph-1
Source
David Bostock (Philosophy of Mathematics [2009], 4.5)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.110
Related Idea
Idea 18106 Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]