If you assume Axiom of Choice, you can define cardinalities in terms of ordinal numbers (an extension of naturals that generalizes the notion of "counting", or

If you assume Axiom of Choice, you can define cardinalities in terms of ordinal numbers (an extension of naturals that generalizes the notion of "counting", or "indexing"). And ordinal numbers are well-ordered, ie. every element has a unique "successor" element.