Ordinal Number

Definition

An ordinal number (or ordinal) is a mathematical object that extends the notion of natural number to represent the order type of well-ordered sets. Ordinals generalize the concept of “position” or “rank” in a sequence.

Construction in Set Theory

In Zermelo-Fraenkel set theory, ordinals are typically defined using von Neumann’s construction:

• The empty set $\varnothing$ represents $0$
• The successor of an ordinal $\alpha$ is $\alpha \cup \{\alpha \}$
• A limit ordinal is the union of all smaller ordinals

This gives:

• $0=\varnothing$
• $1=\{0\}=\{\varnothing \}$
• $2=\{0,1\}=\{\varnothing ,\{\varnothing \}\}$
• $\omega =\{0,1,2,\dots \}$ (first infinite ordinal)

Properties

• Every ordinal is the set of all smaller ordinals
• Ordinals are well-ordered by the membership relation $\in$
• Every well-ordered set is order-isomorphic to a unique ordinal
• Ordinals extend beyond the natural numbers to transfinite ordinals

Types of Ordinals

• Finite ordinals: Correspond to natural numbers $0,1,2,\dots$
• Limit ordinals: Ordinals with no immediate predecessor (e.g., $\omega ,\omega \cdot 2,{\omega }^{\omega }$)
• Successor ordinals: Ordinals of the form $\alpha +1$

Ordinal Arithmetic

Ordinals support arithmetic operations:

• Addition: $\alpha +\beta$
• Multiplication: $\alpha \cdot \beta$
• Exponentiation: ${\alpha }^{\beta }$

Note that ordinal arithmetic is not commutative (e.g., $1+\omega =\omega$ but $\omega +1>\omega$).

Related Concepts

• Cardinal Number: Measures of size rather than order
• Natural Number: Finite ordinals
• Well-Ordering: The structure that ordinals represent
• Set Theory: The framework for defining ordinals
• Transfinite Induction: Proof technique using ordinals