In set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers.
The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality (a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above; in the style of Hilbert's Hotel Infinity).
Using the von Neumann cardinal assignment and the axiom of choice (AC), this successor operation is easy to define: for a cardinal number κ we have where ON is the class of ordinals.
Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals.
The standard definition above is restricted to the case when the cardinal can be well-ordered, i.e. is finite or an aleph.