# Definition:Finite Ordinal

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## 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