Full Idea
Ordinals play the central role in set theory, providing the paradigmatic well-orderings.
Gist of Idea
Ordinals play the central role in set theory, providing the model of well-ordering
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Book Reference
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.89
A Reaction
When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level.