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Full Idea
The ordinal representing the whole series must be different from what represents a segment of itself, with no immediate predecessor, since the series has no last term. ω names the class progression, or generating relation of series of this class.
Gist of Idea
ω names the whole series, or the generating relation of the series of ordinal numbers
Source
Bertrand Russell (The Principles of Mathematics [1903], §291)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.313
A Reaction
He is paraphrasing Cantor's original account of ω.
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 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] |