Definition:Finite Ordinal

Definition

Let $\alpha$ be an ordinal.

Then $\alpha$ is said to be finite if and only if one of the following holds:

$\alpha = \O$

$\alpha = \beta^+$ for some finite ordinal $\beta$

where $\O$ denotes the empty set, and $\beta^+$ is the successor ordinal of $\beta$.

Also known as

In many sources oriented towards set theory, finite ordinals are referred to as natural numbers.

The relation with the natural numbers arises from the multiple definitions of minimally inductive set $\omega$, combined with the definition of $\N$ as $\omega$.

However, in an effort to keep separated the familiar properties of $\N$ and those of finite ordinals, $\mathsf{Pr} \infty \mathsf{fWiki}$ does not identify these intuitively distinct concepts.

Also see

• Definition:Finite Set, which through the von Neumann construction of the natural numbers would be circular if used here.
• Definition:Transfinite Ordinal

• Results about finite ordinals can be found here.

Sources

• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Natural and Ordinal Numbers