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Full Idea
The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered.
Gist of Idea
Ordinals are basic to Cantor's transfinite, to count the sets
Source
Shaughan Lavine (Understanding the Infinite [1994], III.4)
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.54
A Reaction
Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'!
Related Idea
Idea 15916 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]