The cardinality of a finite set is a natural number. The smallest cardinality is 0. The empty set has cardinality 0. If the cardinality of the set "A" is "n", then there is a "next larger" set with cardinality "n"+1 (for example, the set "A" ∪ {"A"}). If ||"A"|| ≤ ||"B"|| ≤ ||"A" ∪ {"A"}||, then either ||"B"|| = ||"A"|| or ||"B"|| = ||"A" ∪ {"A"}||.) There is no largest finite cardinality. 