Cardinal numbers are identified with the set of ordinals preceding them

Question

Here is a description of cardinal number.

Cardinal numbers are identified with the set of ordinals preceding them.

Is this OK?

Answer 1

It would be better to say:

Ordinals are von Neumann ordinals: each ordinal is the set of its predecessors, and cardinal numbers are identified with initial ordinals.

Answer 2

Your statement is unclear, or even wrong, because every ordinal is identified as the set of preceding ordinals. It is also unclear whether or not you mean that every cardinal number is identified with an ordinal, or that every ordinal which can be identified with the set of its predecessors is a cardinal.

Cardinal numbers are identified with initial ordinals, which are ordinals that there is no injection from them into a smaller ordinal.