Questions: Ordinal Numbers: Definition and Order Structure

Question 1

Why does the set {..., −2, −1, 0} (all non-positive integers in their usual order) have no associated ordinal number?

• A. It is an infinite set, and ordinals only apply to finite sets
• B. It is not well-ordered: there is no least element (no smallest non-positive integer), so the set lacks the structural property ordinals require
• C. It uses negative numbers, which cannot be elements of ordinals
• D. Its cardinality is uncountable, placing it beyond the ordinal hierarchy

Answer: (B)

Ordinals are canonical representatives of well-orderings — orderings in which every nonempty subset has a least element. The non-positive integers {..., −2, −1, 0} have no least element: for any n ≤ 0, there exists n−1 < n. Since the set is not well-ordered, it has no ordinal. This is not about finiteness or cardinality — it's about order structure. Well-orderedness is precisely the property that makes transfinite induction work.

Question 2

Two sets are both countably infinite: ω (natural numbers 0, 1, 2, ... in usual order) and ω + 1 (natural numbers followed by one additional element at the end). Which statement is correct?

• A. They are the same ordinal because both are countably infinite — ordinals measure size
• B. ω + 1 ≠ ω: adding a last element changes the order type even though both sets have the same cardinality; ordinals capture order structure, not just size
• C. ω + 1 = ω because infinity plus one is still infinity
• D. ω + 1 must be uncountable since it is strictly larger than ω

Answer: (B)

This is the central insight about ordinals: they capture order type, not cardinality. ω has no last element; ω + 1 has a last element (ω itself). These are structurally different well-orderings, so they are different ordinals. Both are countably infinite — same cardinality — but distinct ordinal numbers. The intuition that 'infinity + 1 = infinity' confuses cardinal arithmetic (where ℵ₀ + 1 = ℵ₀) with ordinal arithmetic (where ω + 1 ≠ ω).

Question 3

In von Neumann's construction, ordinals are defined as sets where the ordering relation 'n < m' is equivalent to set membership 'n ∈ m', eliminating the need for a separate definition of 'less than.'

• T. True
• F. False

Answer: True

This is the elegant economy of von Neumann ordinals: 0 = ∅, 1 = {0}, 2 = {0, 1}, and n < m simply means n ∈ m. Every ordinal is the set of all smaller ordinals, so the 'less than' relation is built into the set structure itself. This is not a coincidence — it is a deliberate design choice that makes ordinals self-contained mathematical objects with their order encoded internally rather than added on separately.

Question 4

Two sets with the same cardinality generally have the same ordinal number, because ordinals and cardinals both measure the 'size' of sets.

• T. True
• F. False

Answer: False

Ordinals capture order type; cardinals capture size. The sets {0, 1, 2, ...} (order type ω) and {0, 1, 2, ..., ω} (order type ω + 1) have the same cardinality (both countably infinite) but different ordinals. Even for finite sets, every 3-element set has the same cardinality, but {a, b, c} ordered differently gives different order types — though for finite well-orderings, cardinality and ordinal happen to coincide. The distinction matters crucially in the transfinite setting.

Question 5

Explain why ordinals capture 'order type' rather than just 'size,' and give a concrete example of two infinite sets that have the same cardinality but different ordinal structure.

Model answer: Order type describes the pattern of how elements are arranged — specifically, whether a well-ordering has a first element, a last element, limit points, etc. Cardinality only counts how many elements there are. Example: ω = {0, 1, 2, ...} and ω + 1 = {0, 1, 2, ..., ω} are both countably infinite (same cardinality) but ω has no last element while ω + 1 does — they are structurally different well-orderings with different ordinals.

Every well-ordered set is order-isomorphic to exactly one ordinal — this uniqueness makes ordinals the canonical yardstick for well-orderings. Two well-orderings that look the same structurally (you can match them element-by-element in an order-preserving way) get the same ordinal. Two that don't match structurally — even if they have the same number of elements — get different ordinals. This is why transfinite induction needs ordinals, not just cardinals: the induction step depends on having a well-defined 'previous' element, which is an order property, not a size property.