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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]
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 10034 | The number of natural numbers is not a natural number [Frege, by George/Velleman] |
| 14143 | ω names the whole series, or the generating relation of the series of ordinal numbers [Russell] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |