Ordinal Number Definition

Set Theory · Axiom Academy

The Von Neumann construction: ordinals as transitive sets well-ordered by membership An ordinal is defined as a transitive set that is well-ordered by the membership relation ∈. What does transitive mean? A set is transitive if all elements of its elements are also its elements. In other words, membership chains collapse: if b is in a and a is in α, then b is also in α. Let's construct the first ordinals step by step: 0 = ∅ - The empty set is the first ordinal 1 = ∅ = 0 - The set containing only 0 2 = ∅, ∅ = 0, 1 - The set containing 0 and 1 3 = ∅, ∅ , ∅, ∅ = 0, 1, 2 - The set containing 0, 1, and 2 n = 0, 1, 2, ..., n-1 - Each natural number n Notice the beautiful pattern: each ordinal contains exactly the ordinals that came before it! 3. Every Ordinal is Its Predecessors The most fundamental property of ordinals: each ordinal equals the set of all smaller ordinals. The ordinal 5 literally is the set 0, 1, 2, 3, 4 Being "less than α" is the same as being "a member of α" The ordering relation < coincides with the membership relation ∈ 4. Ordinals Are Well-Ordered by ∈ The membership relation ∈ provides a natural well-ordering on ordinals. For ordinals α and β, exactly one of the following holds: Why is this well-ordered? Every non-empty set of ordinals has a least element. This is because ordinals are designed to be "downward closed" - if you have an ordinal, you have all the ordinals below it. Ordinals form an infinite hierarchy extending beyond the natural numbers.

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