Iterestingly in the modern approach to set theory there still are both ordinal and cardinal numbers, but you can't say that the cardinal n is the set of all sets with cardinality n, because this is a proper class. Instead we choose a particular representative for this class, which also happens to be well ordered (this is in ZFC, you can still pick representatives for equipotence classes in ZF via Scott's trick but they are not necessarily well ordered or well orderable).