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 Reference
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.313
A Reaction
He is paraphrasing Cantor's original account of ω.