Questions: Von Neumann Ordinals

Question 1

According to the von Neumann construction, what is the ordinal 3?

• A. {1, 2, 3}
• B. {0, 1, 2}
• C. {{∅}}
• D. The set containing only ∅

Answer: (B)

In the von Neumann representation, each ordinal equals the set of all smaller ordinals. So 3 = {0, 1, 2}. It is not {1,2,3} — ordinals begin at 0 = ∅. The ordinal n always has exactly n elements, confirming the size-equals-value property.

Question 2

Ordinal addition is commutative: for any ordinals α and β, α + β = β + α.

• T. True
• F. False

Answer: False

Ordinal arithmetic is not commutative. The standard counterexample: 1 + ω = ω, because appending one element before the sequence 0,1,2,... still yields an ω-sequence. But ω + 1 > ω, because appending one element after the ω-sequence creates a new limit with a successor at the end. Order of concatenation changes the order type.

Question 3

What does it mean for a set α to be transitive, and why is transitivity part of the definition of a von Neumann ordinal?

Model answer: A set α is transitive if every element of α is also a subset of α (x ∈ α implies x ⊆ α). Transitivity ensures each ordinal is closed downward — it contains all elements of its elements — making ordinals canonical, self-contained representations of all smaller ordinals.

Without transitivity, two well-ordered sets with the same order type could look different internally. Transitivity, combined with well-ordering by ∈, forces a unique canonical form: the ordinal α is exactly the set of all ordinals less than α, with no extraneous elements.