Cardinality is a measure of size, and (by definition) all countably infinite sets have the same number of elements based on the principle that any two sets for

Cardinality is a measure of size, and (by definition) all countably infinite sets have the same number of elements based on the principle that any two sets for which we have a bijection between them are of the same size. Ordinals deal with something else: complexity of orderings upon sets. As the ordinals get "larger", the orderings they represent get increasingly complicated. Once we're past the Church-Kleene ordinal, they're so convoluted that we can't algorithmically compare them.