Full Idea
There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements.
Gist of Idea
Two infinite ordinals can represent a single infinite cardinal
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
This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different.