How do ordinal numbers differ from cardinal numbers?

In set theory, cardinal and ordinal numbers serve distinct purposes regarding the properties of sets:

• Cardinal Numbers: These are used to designate the size or "number of elements" of a set. Two sets have the same cardinality if they can be placed into a one-to-one correspondence with each other.12
• Ordinal Numbers: These are used to determine the specific position of elements within a prescribed well-ordered sequence. While cardinal numbers focus on quantity, ordinal numbers focus on order.23

In the Zermelo-Fraenkel set theory (ZFC) system, these concepts are formally linked: cardinal numbers are defined as specific types of ordinal numbers. Specifically, a cardinal number is defined as the least ordinal number to which a given set is equivalent.2

While the arithmetic of finite cardinals and ordinals coincides with that of natural numbers, their behaviors diverge significantly when dealing with infinite (transfinite) sets. The arithmetic of infinite ordinals is complex and varied, whereas the arithmetic of infinite cardinals is often simplified by the axiom of choice, which dictates that the sum or product of two infinite cardinals is simply the maximum of the two.2