Full Idea
It is the usual procedure these days to identify a cardinal number with the earliest ordinal number that has that number of predecessors.
Gist of Idea
A cardinal is the earliest ordinal that has that number of predecessors
Source
David Bostock (Philosophy of Mathematics [2009], 4.5)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.111
A Reaction
This sounds circular, since you need to know the cardinal in order to decide which ordinal is the one you want, but, hey, what do I know?