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Full Idea
Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence).
Gist of Idea
Ordinals are generated by endless succession, followed by a limit ordinal
Source
report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.52