Ordinals already have an order imposed on them. When you write (2, 3, 4, ...., 0, 1) you are imposing an order and in this particular order 0 has no immediate

Ordinals already have an order imposed on them. When you write (2, 3, 4, ...., 0, 1) you are imposing an order and in this particular order 0 has no immediate predecessor. This is key. 0 is the only element with predecessors but no immediate predecessor. The standard ordering of the natural numbers doesn't have this property. The cardinality of a set is independent of any ordering you put on it. I hope this helps to distinguish between these concepts. Omega+1 has the same cardinality as Omega but different ordinality. It is true though that a set is uncountable (cardinal concept) if and only if every ordering of it admits elements with predecessors but no immediate predecessor.