Are cardinal numbers sets in ZFC?

Question

Are cardinal numbers sets in ZFC, or just proper classes? If they are sets, what is their structure?

Answer 1

Yes. Since every set can be well ordered, we identify the cardinal numbers with specific ordinals. Namely a set is a cardinal number if and only if it is an ordinal (transitive set and we'll ordered by $\in$) and no smaller ordinal is equipotent with it.

Answer 2

They are sets. Usual definition is that cardinal numbers are least ordinal numbers of a given cardinality. Therefore, each cardinal numbers is also a ordinal, so their structure depend on how you define ordinals.

Answer 3

Cardinalities ($>0$) are not sets if you define them as equivalence classes under bijectivity. But thanks to the C in ZFC, each of these classes contains at least one ordinal, and hence a smallest ordinal. So if you define cardinalities as ordinals that cannot be bijected to a proper subset, then cardinalities are sets.